LICRARY 

UNIVCK3ITY  Of 
CALIFORNIA, 

9AN  DIEGO 


THE  FOUNDATIONS  OF  MUSIC 


CAMBRIDGE  UNIVERSITY  PRESS 

C.  F.  CLAY,  Manager 

LONDON  :  FETTER  LANE,  E.G.  4 

NEW  YORK    :    G.P.PUTNAM'S  SONS 

BOMBAY       •) 

CALCUTTA  {•  MACMILLAN  AND  CO.,  Ltd. 

MADRAS       J 

TORONTO  :  J.  M.  DENT  AND  SONS,  Ltd. 

TOKYO :  MARUZEN-KABUSHIKI-KAISHA 

ALL   RIGHTS   RESERVED 


^390  "^ 


THE 

FOUNDATIONS  OF  MUSIC 


BY 


HENRY  J.  WATT,  D.Phil. 

Author  of  The  Psychology  oj  Sound 

Lecturer  on  Psychology  in  the  University  of  Glasgow  and  to  the 
Glasgow  Provincial  Committee  for  the  Training  of  Teachers. 
Sometime  Lecturer  on  Psychology  in  the  University  of  Liverpool. 


CAMBRIDGE 

AT  THE  UNIVERSITY  PRESS 

1919 


THESE  WORKS 
ON   SOUND  AND   ON   MUSIC 

I  DEDICATE 
TO  MY  WIFE  AND  HER  ART 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/foundationsofmusOOwattiala 


PREFACE 

In  my  previous  volume  The  Psychology  of  Sound  I  made  a  minutely 
critical  analysis  of  the  elementary  phenomena  of  sound  and  their 
simpler  complexities,  and  I  developed  what  seemed  to  me  to  be  the 
only  systematically  true  and  promising  theory  of  these  phenomena. 
The  work  was  necessarily  addressed  to  those  who  are  primarily  interested 
in  such  a  study,  i.e.  to  psychologists  and  to  physiologists.  But  I 
endeavoured  to  make  the  material  as  interesting  to  the  theoretical 
musician  as  was  possible  under  the  circumstances. 

Not  that  the  latter  has  little  interest  in  such  fundamental  analysis. 
On  the  contrary  he  is  profoundly  concerned  to  know  how  his  art  springs 
from  its  roots  in  mere  sound  and  to  see  that  the  foundations  ascribed 
to  it  are  such  as  will  evidently  suffice  to  bear  the  whole  superstructure 
of  music.  But  the  purely  psychological  or  'phenomenal'  point  of  view 
could  not  but  be  new  and  strange  to  his  mind,  requiring  some  time  to 
come  into  growth  and  fruition  there.  Once  the  essential  nature  of  the 
position  has  been  grasped,  its  spontaneous  development  is  certain. 

There  is  no  inherent  difficulty  in  ascribing  volume  and  order  to 
sounds  or  to  tones.  The  difficulty  springs  merely  from  the  unfamiliarity 
of  the  object  in  such  connexions.  At  the  present  day  conviction  is 
much  more  easily  secured  for  descriptions  and  theories  of  material 
objects — even  although  many  of  their  students  may  never  have  come 
into  contact  with  them  at  all — than  it  is  for  descriptions  and  theories 
of  psychical  objects,  although  their  students  are  almost  of  necessity 
constantly  face  to  face  with  them  at  any  desired  moment.  Every  one 
who  takes  any  interest  in  music  has  had  unlimited  opportunities  of 
turning  his  observations  upon  tones  and  their  sequences  and  combina- 
tions. But  in  the  great  majority  of  cases  he  has  seldom,  if  ever,  looked 
studiously  at  pictures  or  models  of  the  sensory  organ  of  hearing  and  in 
all  probability  knows  nothing  of  that  organ  by  direct  observation  of  it. 
But  he  will  nevertheless  drink  in  a  description  and  theory  of  the  material 
organ  with  avidity,  while  he  will  turn  a  bored  and  sceptical  ear  to  a 
direct  analysis  and  theory  of  tones,  although  for  both  purposes  similar 
methods  and  explanations  may  have  been  used.  The  mere  postulation 
of  a  material  thing  as  the  bearer  of  volumes  and  orders  and  their 
coincidences  and  overlappings  seems  to  bring  a  special  comfort  to  the 
mind.  • 


viu  PREFACE 

It  would  be  wrong  to  suggest  that  this  scepticism  is  quite  general 
in  it«  scope.  .The  artist  is  certainly  clearly  aware  that  what  he  usually 
judges  and  accepts  or  rejects  is  the  direct  phenomenal  impression  that 
is  immediately  before  his  mind's  gaze.  But  when  it  comes  to  science 
and  to  theory,  what  he  has  learnt  to  crave  for  is  in  the  main  a  material- 
istic exposition.  The  musician  has  long  since  accustomed  himself  to  a 
theoretical  diet  of  beats,  partials,  and  material-mathematical  expressions 
for  intervals.  Such  things  seem  real  and  tangible,  as  it  were.  But, 
after  all,  they  seem  so  in  most  cases  "only  because  they  are  more 
familiar. 

Much  of  the  difficulty  is  due  also  to  a  widespread  shallow  attitude 
towards  any  scientific  aesthetics, — an  attitude  unfortunately  greatly 
encouraged  amongst  musical  theorists  by  Helmholtz's  very  unsatis- 
factory distinction  between  natural  law  and  aesthetic  principles.  The 
mere  existence  and  operation  of  personally  subjective  forces  that  affect 
our  artistic  judgments  seem  to  convince  so  many  people  that  no  science 
of  these  judgments  can  be  achieved.  There  is  no  disputing  about 
tastes,  they  say.  And  if  a  body  of  critical  knowledge  can  be  extracted 
from  established  works  of  fine  art,  such  rules  are  held  to  be  merely 
the  conventions  of  the  ages  that  created  them.  The  next  genius  that 
comes  along  may  blow  the  whole  system  to  the  winds  of  oblivion.  So 
it  seems  to  those  who  are  struck  most  of  all  by  the  innovations  of  each 
master  and  have  not  perhaps  the  patience  to  follow  out  the  great 
purpose  that  is  common  to  them  all  and  that  each  merely  carries  on  to 
finer  and  finer  issues.  But  a  great  master  is  by  no  means  an  accident. 
He  is  one  who,  taking  himself  as  a  man  amongst  many,  has  learned 
how  to  construct  an  enduring  object  that  is  far  more  likely  to  arouse 
in  others  the  joys  of  beauty  he  has  felt  and  anticipated  for  them  than  are 
the  works  of  lesser  minds.  His  medium  is  the  orderly  realm  of  mind  that 
he  shares  with  his  fellows.  But  its  laws  are  not  his  creation;  they  are 
only  his  discovery.  He  has  learnt  to  turn  them  to  his  will.  The  scientist 
who  comes  after  him  has,  with  his  help,  to  formulate  them  in  knowledge. 
That  knowledge  is  the  science  of  aesthetics. 

In  this  volume  I  have  sought  more  or  less  evenly  to  serve  the  purposes 
of  both  the  psychologist  and  the  musician.  In  order  to  make  the  work 
complete  in  itself  up  to  a  certain  point  I  have  traversed  the  ground 
covered  in  the  psychological  part  of  the  earlier  volume,  omitting  only 
those  parts  that  are  of  little  interest  to  the  musician.  All  critical  dis- 
cussion has  been  passed  over  at  this  stage,  so  that  the  earlier  part  of 


PREFACE  ix 

this  volume  is  more  or  less  a  careful  and  straightforward  ('dogmatic') 
exposition  of  the  fundamental  notions  of  the  psychology  of  tone.  I  think 
the  musician  should  find  it  useful  and  helpful,  as  also  will  those  who 
wish  to  have  an  exposition  of  the  system  without  the  technical  dis- 
cussions and  criticisms.  Those  who  are  familiar  with  the  previous 
volume  will  hardly  find  anything  new  before  chapter  ix  (p.  55).  Except 
in  so  far  as  they  are  interested  in  straightforward  and  logical  exposition, 
they  may  begin  the  work  at  that  point.  The  previous  pages,  however, 
are  not  in  any  sense  a  mere  repetition  of  the  earlier  volume.  They 
have  been  written  entirely  afresh.  Only,  as  is  after  all  inevitable,  they 
are  based  on  the  same  body  of  facts  and  notions  as  was  the  analytic 
psychological  work  of  the  earlier  volume.  I  have  not  gone  into  any 
binaural  or  physiological  problems  this  time. 

The  parts  beyond  chapter  ix  are  addressed  both  to  psychologists 
and  to  musicians.  In  the  preface  to  the  previous  volume  I  said  that 
my  "  theoretical  constructions  must  be  carried  somewhat  farther  before 
they  can  be  held  to  have  passed  fully  over  into  the  elements  consciously 
used  [and  known]  by  productive  musicians  and  appreciative  listeners.... 
The  working  musician  definitely  takes  over  at  a  certain  point  the  raw 
materials  of  his  art  from  the  real  psychical  processes  of  hearing, 
inaccessible  in  full  to  observation,  and  then  proceeds  to  construct 
from  them  vast  new  realms  without  consulting  anything  that  lies  beyond 
the  ken  of  observation."  In  this  volume  I  think  I  have  succeeded  in 
carrying  the  psychological  groundwork  of  the  previous  one  forward  so 
as  to  bridge  the  gulf  between  the  psychological  elements  and  processes 
of  music  on  the  one  hand  and  on  the  other  the  sensory  stuff  and  functions 
of  music  as  the  musician  observes  them.  If  ray  results  and  analysis 
are  valid,  the  musician  should  now  have  a  nearly  complete  and  sure 
basis  to  work  upon,  that  will  give  a  scientific  foundation  to  all  his 
elementary  observations  and  satisfy  him  with  a  sense  of  firm  ground 
upon  which  to  build. 

The  work  of  carrying  the  psychological  analysis  thus  far  has  not 
been  light.  For  as  things  have  been  till  now,  the  probability  of  any 
one  person  being  equally  and  fully  conversant  with  the  science  of 
psychology  and  with  musical  history  and  theory  was  exceedingly  small. 
I  am  aware  of  the  great  deficiencies  in  my  own  preparedness  for  the 
latter  half  of  this  great  double  task.  I  have  tried  to  make  up  for  want 
of  experience  by  keeping  closely  in  touch  with  the  general  trend  of 
the  judgments  of  ripened  masters  in  musical  theory  of  the  empirical 
analytical  order.   I  feel  sure  that  results  have  offered  themselves  to  my 


X  PREFACE 

hand  that  theirs  passed  by,  however  closely.  And  I  am  certain  that 
thev  will  not  be  loth  to  recognise  the  validity  and  usefulness  of  these 
results.  I  only  wish  that  they  would  feel  impelled  to  make  themselves 
familiar  enough  with  the  results  of  psychological  work  on  sound  to 
carry  the  new  and  promising  basis  of  theory  far  forwards  into  their 
own  fields.  In  order  to  facilitate  this  junction  and  continuity  of  work 
I  have  ventured  as  far  out  into  musical  regions  as  the  material  accessible 
to  me  will  allow.  Analytic  musicians  will  certainly  be  able  to  carry 
things  much  farther  on  until  the  new  outlook  permeates  the  whole 
theory  of  their  art. 

There  is  still  much  to  be  done.  Although  I  feel  a  growing  assurance 
that  the  lines  of  analysis  I  have  followed  lead  in  the  right  direction, 
I  still  feel,  not  only  that  the  elements  of  analysis  are  not  nearly  well 
enough  assimilated  to  one  another,  but  also  that  a  much  more  detailed 
and  exact  foundation  is  required  for  their  final  acceptance  than  the 
empirical  generalisations  of  writers  on  harmony  afford,  however  correct 
and  worthy  these  may  be.  A  better  form  of  evidence  is  required;  and 
the  best  I  can  imagine  would  be  a  statistical  study  of  the  great  composers, 
seconded  by  systematic  experiments  with  the  best  observers. 

In  order  to  bring  my  results  the  more  quickly  into  touch  with 
practical  issues  and  exposition,  I  have  again  ventured  upon  an  exposition 
of  results  in  the  most  familiar  terms  of  elementary  musical  knowledge. 
The  underlying  idea  of  this  account  is  the  construction  of  a  simple 
framework  round  which  an  introductory  account  of  the  basis  and  rules 
of  harmony  might  be  built. 

Some  text-books  still  maintain  the  effort  to  link  a  system  of  harmony 
to  the  traditional  lines  of  scientific  explanation  founded  upon  the 
harmonic  partial  tones  of  musical  sounds.  Of  this  E.  Front's  splendid 
treatise  is  in  its  earlier  form  a  most  notable  example.  But  the  effort 
has  been  a  complete  failure,  whose  only  result  must  have  been  to 
bewilder  any  student  of  a  logical  turn  of  mind.  Many  other  writers 
have  abandoned  every  sort  of  scientific  introduction;  and  Prout  has 
followed  them  in  the  later  edition  of  his  work.  That  course  seems,  on 
the  whole,  preferable  to  the  former.  But  the  bald  exposition  of  analytic 
generalisations,  however  true  to  the  great  masters  they  may  be,  can 
never  be  enough.  The  mind  craves  for  some  logical  nexus  to  give  the 
whole  mass  spontaneous  life.  Even  analytic  exposition  can  never  be 
complete  until  it  has  developed  into  a  logical  system  whose  fundaments 
bear  the  higher  refinements  as  a  tree  bears  its  fruit.   And  these  refine- 


PREFACE  xi 

ments  can  hardly  be  properly  approached  and  stated  until  the  system 
inherent  in  the  products  of  analysis  has  been  discovered. 

I  hope  the  lines  of  introduction  I  suggest  will  help  to  relieve  the 
beginning  students  whose  musical  mind  is  not  good  enough  to  be  a 
test  for  every  effect  and  a  storehouse  for  every  impression,  of  the 
perplexity  and  confusion  that  so  soon  overwhelms  his  first  efforts  at 
'harmony.'  And  perhaps  the  teacher  will  find  comfort  and  stimulus 
in  having  (I  hope)  a  good  explanation  to  offer,  by  which  the  student's 
logical  mind  may  be  brought  to  the  support  of  his  natural  gifts  of  ear. 

It  may  seem  strange  to  suggest  that  the  exposition  of  the  theory  of 
harmony  has  hitherto  been  devoid  of  system.  There  has  certainly  been 
no  lack  of  desire  for  system  and  of  effort  to  form  it.  Dr  Shirlaw  has 
recently  given  us  a  lengthy  account  of  "the  chief  systems  of  harmony 
from  Rameau  to  the  present  day."  These  are  very  numerous  and  of 
the  greatest  diversity.  But  it  is  evident  from  a  study  of  them  that 
they  are  all  mere  castles  in  the  air,  as  it  were.  They  lack,  one  and  all, 
any  proper  sort  of  foundation.  Each  man's  construction  is  like  a  toy 
castle  built  upon  his  outstretched  hand.  The  parts  are  in  the  main 
merely  laid  down  beside  and  upon  one  another.  The  weight  of  the  upper 
parts  repeatedly  pulls  out  the  joints  from  their  places;  for  they  are 
not  bound  by  any  mortar.  And  the  builder's  own  care  and  anxiety 
does  the  most  to  precipitate  the  fall  of  the  pile.  Perhaps  in  the  end 
he  throws  the  whole  game  into  the  fireplace  and  warms  himself  by  it 
while  he  reflects  on  what  useful  thing  he  might  do  instead. 

Dr  Shirlaw,  it  is  true,  in  spite  of  the  extensive  criticism  he  bestows 
upon  these  theorists,  still  believes  that  a  good  foundation  for  musical 
theory  is  to  be  found  in  the  "resonance  of  the  sonorous  body."  "As 
the  sounds  of  this  harmony  are  contained  in  the  resonance  of  musical 
sound  itself,  all  harmony  has  its  source  in  a  single  musical  sound" 
(60,  481).  But  the  day  is  well  past  in  which  a  system  can  hope  for  a 
moment's  success  that  rests  upon  such  philosophical  naivety.  The 
scepticism  of  Prout  and  others  is  far  more  worthy  and  hopeful. 

TJie  sonorous  body  is  out  of  the  question.  Between  it  and  music 
there  lie  not  only  the  phenomenal  material  of  sense  but  the  mechanisms 
of  the  ear  as  well.  The  ear  must  stand  in  real  systematic  continuity 
with  the  physical  processes  of  sound,  including  as  a  mere  part  the  musical 
sonorous  body;  the  phenomena  of  sense  must  bear  a  similar  relation 
to  the  ear.  The  problems  of  musical  science  consist  primarily  in  showing 
the  system  of  bonds  and  relations  by  which  the  stuff  of  auditory  sense 


xii  PREFACE 

is  built  up  into  music  that  delights  the  soul.  There  are  doubtless  various 
other  ways  of  building  up  structures  of  sound  than  the  musical  one. 
But  the  latter  is  one  of  the  most  important.  In  it  the  aesthetic  test  is 
applied;  a  process  of  selection  is  set  over  nature's  indiscriminating 
generosity.  But  it  is  obvious  that  the  aesthetic  test  no  more  accounts 
for  the  auditory  material  of  music  than  a  five  shilling  entrance  fee 
accounts  for  the  audience  that  appears  at  a  concert. 

As  far  as  the  sensory  stuff  of  music  is  concerned,  then,  and  apart 
from  such  things  as  rhythm  and  musical  form,  no  tenable  theory  either 
of  the  auditory  basis  or  of  the  aesthetics  of  it  has  ever  yet  been  advanced. 
But  there  does  exist  a  considerable  body  of  generalisations  that  have 
been  won  from  the  works  of  the  masters  by  analytical  induction.  By 
this  I  do  not  mean  merely  the  useful  attempts  that  have  been  made 
to  form  systems  of  chords;  for  these  have  been  largely  vitiated  by  the 
attempt  to  derive  the  chords  from  some  fundamental  chord  or  other, 
when  it  was  evident  for  strict  logical  thought  and  intuition  (if  one  may 
say  so),  that  all  the  time  no  satisfying  criterion  existed  by  which  any 
such  fundamental  chord  could  be  established.  D'Alembert  expressed 
this  thought  in  relation  to  Rameau's  efforts  when  he  said  they  yielded 
no  demonstration,  but  only  a  system.  I  mean  rather  the  body  of 
experience  that  has  been  known  as  the  rules  of  part-writing  and  of 
the  formation  of  melody  and  the  factors  that  modify  the  operation 
of  these  rules.  These  rules  have  been  the  step-children  of  musical  theory, 
ignored  and  despised  as  far  as  any  system  building  was  concerned, 
treated  only  as  accidents  of  the  all-absorbing  science  of  chords  and 
their  origins.  But  this  despised  child  of  musical  empiry  may  well 
take  precedence  in  musical  science  and  end  by  being  the  queen  who 
will  show  her  sisters  in  music  what  their  functions  really  are. 

I  am  far  from  suggesting  that  all  the  main  problems  have  been 
solved.  But  it  is  good  that  lines  begin  to  appear  that  seem  promising 
and  strong.  Many  must  labour  at  the  task  before  it  will  ever  be  wrought 
to  the  satisfaction  of  all.  Still,  there  seems  to  be  no  reason  now  why 
rapid  progress  should  not  be  made,  even  if  no  one  would  be  quite  so 
hopeful  as  Kant  was  about  his  efforts  as  to  be  sorry  for  the  next  genera- 
tion that  would  have  nothing  to  do  but  learn  the  results. 

When  one  surveys  the  actual  changes  in  the  fundamental  notions 
of  a  subject  like  music  required  by  even  very  extensive  inductions  and 
analyses,  they  may  seem  to  be  surprisingly  small.  This  is  perhaps 
most  striking  in  the  problems  of  consonance  and  dissonance.  It  is  not 
so  much  a  striking  change  of  substance  that  is  produced  but  a  far- 


PREFACE  xiii 

reaching  improvement  in  the  body-building  capacities  of  the  fundamental 
nucleus.  As  cytology  shows  us,  trifling  changes  in  the  numbers  and 
arrangement  of  chromosomes  may  alter  the  resulting  organism  pro- 
foundly. And  ages  are  required  to  attain  these  trifling  changes.  So  in 
science  :  some  fundamental  notions  are  lethal,  they  can  compound  to 
no  living  organism;  others  very  slightly  different  grow  and  reproduce 
with  great  vigour.  That  I  hope  may  be  true  of  my  own  analyses.  And 
as  great  interest  attaches  to  the  way  in  which  new  notions  of  promise 
come  to  light,  I  have  not  sought  to  obliterate  the  traces  of  this  process 
in  my  writing.  A  straightforward  factual  and  logical  development 
naturally  expunges  all  these  things.  But  that  method  really  takes 
much  for  granted  and  is  only  to  be  used  when  men  are  already  well 
disposed  to  be  convinced  by  the  established  ways  of  a  science. 

I  am  indebted  to  Prof.  W.  B.  Stevenson  for  a  summary  of 
de  Pearsall's  pamphlet  in  the  British  Museum,  and  to  my  brother, 
Rev.  T.  M.  Watt,  for  reading  the  proofs. 

H.  J.  W. 

30th  March,  1918. 


CONTENTS 

PAGE 

Preface  . vii 

CHAP. 

I.  The  reduction  of  instrumental  tones  to  a  single  series  of  pure  tones  1 

II.  Analytic  description  and  theory  of  the  series  of  pure  tones  .        .  6 

III.  Degrees  and  theory  of  consonance  and  dissonance  (fusion)      .        .  15 

IV.  The  relations  of  fusion  to  beats,  partials,  and  difference-tones    .        .  23 

V.  The  consonance  of  successive  tones 31 

VI.  The  nature  of  interval 36 

VII.  The  musical  range  of  pitch 45 

VIII.  Our  point  of  view  towards  the  auditory  field          ....  48 
IX.      The  relative  importance  of  synthesis  and  of  analysis       ...  55 

X.  The  equivalence  of  octaves 65 

XI.  Consecutive  fifths 81 

XII.  The  system  of  facts  regarding  consecutives 98 

Xin.    The  reason  for  the  prohibition  of  consecutives        .        .        .        .  109 
XrV.    Exceptions  to  the  prohibition  of  consecutives          .        .        .        .115 

XV.  Hidden  octaves  and  fifths,  etc 122 

XVI.  A  foiurth  from  the  bass 133 

XVII.  Common  chords  or  concordance 141 

XVIII.  Melodic  motion  in  relation  to  degrees  of  consonance      .        .        .  151 
XIX.    Melody  (or  paraphony)  as  the  primary  basis  of  music    .        .        .160 

XX.  The  factors  that  modify  paraphony 169 

XXI.  Retrospect  and  the  outlook  for  theory 185 

XXII.  Synopsis  or  outhnes  of  instruction 198 

XXIII.  The  objectivity  of  beauty 214 

XXIV.  Aesthetics  as  a  pure  science 221 

Works  cited 233 

Index  of  Authors 236 

Index  of  Subjects 238 


We  shall  now  proceed  to  the  consideration  of  Harmonic  and  its  parts.  It  is  to  be 
observed  that  in  general  the  subject  of  our  study  is  the  question:  In  melody  of  every 
kind  what  are  the  natural  laws  according  to  which  the  voice  in  ascending  or  descending 
places  the  intervals?  For  we  hold  that  the  voice  follows  a  natural  law  in  its  motion, 
and  does  not  place  the  intervals  at  random.  And  of  our  answers  we  endeavour  to 
supply  proofs  that  will  be  in  agreement  with  the  phenomena — in  this  unUke  our 
predecessors.  For  some  of  these  introduced  extraneous  reasoning,  and,  rejecting 
the  senses  as  inaccurate,  fabricated  rational  principles,  asserting  that  height  and 
depth  of  pitch  consist  in  certain  numerical  ratios  and  relative  rates  of  vibration — a 
theory  utterly  extraneous  to  the  subject  and  quite  at  variance  with  the  phenomena: 
while  others,  dispensing  with  reason  and  demonstration,  confined  themselves  to 
isolat-ed  dogmatic  statements,  not  being  successful  either  in  their  enumeration  of 
the  mere  phenomena.  It  is  our  endeavour  that  the  principles  which  we  assume 
shall  without  exception  be  evident  to  those  who  understand  music,  and  that  we 
shall  advance  to  our  conclusions  by  strict  demonstration. 

AbISTOXENUS  (cf.  1,  I88f.), 

Symphonic  are  those  in  which,  when  they  are  simultaneously  struck  or  blown  on  the 
flute,  the  melos  of  the  lower  in  relation  to  the  higher  or  conversely  is  always  the 
same  or  (in  which),  as  it  were,  a  fusion  in  the  performance  of  two  tones  occurs  and 
a  kind  of  unity  results. 

Diaphonic  are  those  in  which,  when  they  are  simultaneously  struck  or  blown,  nothing 
of  the  melos  of  the  lower  in  relation  to  the  higher  or  conversely  appears  to  be  the 
same  or  which  show  no  sort  of  fusion  in  relation  to  one  another. 

Paraphonic  are  those  that,  standing  in  the  middle  between  the  symphonic  and  the 
diaphonic,  yet  appear  symphonic  when  played  [on  instruments,  or  "in  heterophonic 
passages  on  inatrnments  "  (11,  139)]. 

Gaudentius  (cf.  54, 69). 


CHAPTER  I 

THE  REDUCTION  OF  INSTRUMENTAL  TONES  TO  A  SINGLE 
SERIES  OF  PURE  TONES 

The  world  of  sound  is  bounded  by  the  two  extremes  of  pure  tone  and 
mere  noise.  The  home  of  music  lies  in  the  lands  around  the  ideal  of 
tone.   This  ideal  forms  the  first  problem  of  musical  science. 

It  is  properly  termed  an  ideal  because  pure  tones  rarely,  if  ever, 
occur  under  natural  circumstances.  That  is  evident  from  the  familiar 
fact  that,  however  perfectly  a  musical  instrument  may  be  played,  its 
tones  are  easily  distinguishable  from  those  of  other  instruments,  even 
though  they  may  be  of  the  same  pitch.  The  same  series  of  tones  of 
exactly  the  same  pitches,  e.g.  the  diatonic  scale  on  a  c^  of  264  vibrations 
per  second,  may  be  given  by  an  indefinite  number  of  instruments,  and 
will  be  recognised  as  difEerent  on  each,  in  spite  of  the  sameness  that  is 
obviously  common  to  all. 

This  peculiar  complexity  of  tone  has  been  explained  by  modern 
research  in  a  way  that  at  least  in  principle  is  complete  and  final.  How- 
ever pure  and  beautiful  an  instrumental  tone  may  be,  it  can  be  analysed 
into  audible  parts  by  special  means  of  two  kinds.  In  the  first  the  ear 
is  provided  with  instruments  which  wiU  increase  the  intensity  of  certain 
parts  of  the  tone — if  they  are  present  in  the  tone  to  be  analysed — 
beyond  that  of  the  other  parts.  When  the  resonator  is  placed  against 
the  ear,  it  seems  to  be  full  of  the  magnified  sound,  whose  pitch  may 
be  surprisingly  different  from  that  of  the  tone  it  comes  from.  But  its 
presence  in  the  resonator  is  easily  shown  to  depend  upon  the  studied 
tone.  Whenever  the  resonator  is  placed  against  the  ear,  it  appears; 
and  it  only  appears  in  the  resonator  when  the  tone  being  analysed  is 
sounded,  unless,  of  course,  it  is  given  from  some  other  source  at  the 
same  time.  Any  such  experimental  error  can  be  easily  avoided  in  most 
cases.  After  a  little  practice  with  the  resonator,  the  partial  tone  will 
often  be  distinguishable  in  the  whole  tone  even  when  the  resonator  has 
not  been  placed  upon  the  ear.  This  is  not  the  result  of  imagination  or 
illusion.  It  only  means  that  the  ear  has  now  been  trained  for  this 
particular  case  to  expect  a  certain  partial  tone  and  to  direct  observation 

W.  F.  M.  1 


2  REDUCTION  OF  INSTRUMENTAL  TONES  [ch. 

specially  upon  it,  so  that  it  appears  to  be  more  or  less  abstracted  from 
the  whole. 

A  generalisation  of  this  procedure  gives  the  second  method  of  analysis. 
The  ear  is  first  prepared  for  special  observation  (or  abstraction)  by 
listening  for  some  time  to  the  tone  expected  to  occur.  The  tone  to  be 
analysed  is  then  presented  and  if  it  contains  the  prepared  tone  as  a 
partial,  the  latter  will  probably  be  heard  sounding  faintly  through  the 
whole.  If  the  ear  has  been  prepared  for  a  tone  whose  pitch  is  not  quite 
the  same  as  that  of  the  partial  actually  present  in  the  whole,  the  listener 
will  not  hear  the  pit«h  he  expects,  but  another  that  lies  in  pitch  near  the 
one  expected.  When  the  ear  has  thus  been  trained  for  many  or  for  all 
the  partials  of  a  tone,  it  may  be  able  to  run  through  them  all  in  sequence 
without  any  special  preparation.  And  in  the  course  of  time  it  may  learn 
to  do  this  for  any  sound  of  a  tonal  nature.  Even  then,  however,  the 
tones  of  well  played  musical  instruments  do  not  cease  to  be  the 
beautifully  perfect  unities  they  were  before.  They  do  not  fall  to  pieces, 
as  it  were,  permanently,  but  only  when  the  attention  is  concentrated 
and  moved  from  one  of  the  partials  to  another. 

The  occurrence  of  these  partials  is  due  to  the  fact  that  most  musical 
instruments  when  brought  into  a  certain  rate  of  vibration — n  times 
per  second, — fall  at  the  same  time  into  various  rates  of  vibrations  that 
may  be  any  whole  multiple  of  n  :  2n,  3n,  An,  5n,  etc.  The  pitches 
corresponding  to  these  ratios  of  vibration  are  :  octave,  octave  and  fifth, 
double  octave,  double  octave  and  third,  double  octave  and  fifth  (6w),  a 
pitch  shghtly  flatter  than  the  iP  above  {In),  the  triple  octave  (8n),  then 
the  d  (9w)  and  the  e  (lOn)  above  this  c,  and  so  on.  The  pitch  of  any 
partial  may  easily  be  reckoned  out  from  a  knowledge  of  the  ratios  for 
the  chromatic  scale  and  by  approximations  thereto.    These  ratios  are  : 

c,     d>,      d,       ^,       e,       f,      p,     g,       aP,      a,      Jp, 
24,  27,  30,     32,  36,  40, 


d. 

^, 

e. 

/, 

P, 

0, 

27, 

30, 

32, 

36, 

f 

6 
6 

f 

1 

If 

f 

b, 

ci 

45, 

48 

¥ 

f 

116.965  4  45  3  8  5__ 

ti5B  6?  t  TZ¥  "^  ^~r 
All  of  these  ratios  can  be  derived  from  those  of  the  octave  (2  :  1), 
fifth  (3  :  2),  and  major  third  (5  :  4)  that  have  played  so  important  a 
part  in  the  history  of  musical  theory. 

The  existence  of  partial  tones  has  been  confirmed  objectively  in  a 
number  of  ways.  In  various  cases  the  presence  of  vibrations  corre- 
sponding to  the  pitch  of  the  partials  heard  can  be  demonstrated  to 
vision.  A  long  stretched  string  may  be  seen  to  vibrate,  not  only  in  its 
whole  length,  but  also  in  parts  of  such  length  as  would  give  the  various 


I]  TO  A  SINGLE  SERIES  OF  PURE  TONES  3 

partials  as  independent  tones,  if  these  parts  of  the  string  were  made  to 
vibrate  separately  from  the  rest  of  the  string  (cf,  35,  9if.).  In  other 
cases  the  motion  of  a  minute  mirror  standing  in  connexion  with  a 
vibrating  membrane  may  be  photographed  with  so  little  error  that  the 
result  may  be  taken  as  representing  the  motions  of  the  air  that  excite 
the  vibrating  membrane  (40,  78  s.).  (Phonography  is  dependent  upon 
such  a  vibrating  membrane,  and  everyone  is  aware  how  good  a  repro- 
duction of  sound  may  be  obtained  thereby.)  When  the  photographs  so 
obtained  are  subjected  to  mechanical  'harmonic  analysis,'  the  partial 
tones  that  result  correspond  very  closely  with  those  that  can  be  heard 
by  the  most  careful  analysis  with  resonators  or  with  prepared  attention. 

Such  studies  as  these  show  that  tones  of  the  same  nominal  pitch 
from  different  instruments  differ  only  in  respect  of  the  group  of  partials 
from  the  full  series  (of  possible  multiples  of  n)  that  they  contain  and  in 
the  relative  strength  of  these.  Some  tones  like  those  of  tuning-forks 
and  of  the  flute  contain  very  few  partials,  perhaps  only  the  first.  Others, 
such  as  pianoforte  tones,  are  rich  in  the  lower  partials.  Others  again, 
like  those  of  the  trumpets,  contain  a  host  of  high  partials  in  great 
strength,  which  give  them  their  peculiar  brightness  and  brilliancy.  And 
so  on.  The  results  of  this  line  of  study  will  be  found  extensively  in 
special  treatises  and  in  text-books  of  physics  (40,  175 ff.;  20,  ii8f.,  etc.). 

Partials  may  be  eliminated  from  the  tones  of  any  instrument  by 
the  method  of  physical  interference.  That  consists  in  principle  of  the 
conduction  of  a  sound  containing  at  least  one  partial  other  than  the 
fundamental  component  of  the  tone,  along  a  tube  which  for  a  certain 
length  is  double  and  then  unites  again  to  enter  the  ear.  The  one  doubled 
part  is  made  longer  than  the  other  by  half  a  wave-length  of  the  partial 
to  be  eliminated.  When  the  parts  unite,  each  will  bring  this  partial  in 
exactly  opposite  phase  to  the  other.  If  the  one  is  at  the  phase  of 
maximal  condensation  of  the  air,  the  other  will  be  at  that  of  maximal 
rarefaction;  and  the  result  will  be  the  elimination  of  that  component 
of  the  aerial  disturbance.  If  there  are  many  partials  in  the  tone,  there 
will  have  to  be  a  special  device  of  this  kind  for  the  elimination  of  each, 
so  that  the  apparatus  for  the  production  of  a  pure  tone  from  an  instru- 
mental one  is  apt  to  be  somewhat  complicated,  unless  special  care  is 
taken  to  begin  with  a  tone  containing  very  few  partials,  such  as  that 
of  a  tuning-fork. 

In  spite  of  the  difficulties  that  thus  face  any  complete  generalisation, 
no  one  doubts  for  a  moment  that  the  series  of  perfectly  pure  tones,  each 
consisting  of  a  fundamental  with  no  upper  partials  at  all,  thus  isolated, 

1—2 


4  REDUCTION  OF  INSTRUMENTAL  TONES  [ch.  i 

is  one  and  the  same,  no  matter  from  what  instrument  it  may  have 
been  derived.  This  is  in  perfect  accordance  with  the  results  of  the 
attentional  analysis  of  instrumental  tones.  For  the  partials  separated 
by  the  attention  do  not  seem  to  differ  from  the  (pure)  tones  of  identical 
pitch,  otherwise  produced,  in  any  such  way  as  would  make  us  believe 
that  the  series  of  pure  tones  is  not  the  same  whatever  its  source  may  be. 
Thus  we  obtain  a  simpler  starting-point  for  our  study  of  tone — the 
series  of  pure  tones,  each  one  of  which  corresponds  to  a  certain  fixed 
rate  of  aerial  vibrations,  unmixed  with  any  other  rate.  Now  this  series 
is  perfectly  continuous.  If  we  start  from  any  ordinary  pitch,  e.g.  middle  c 
of  264  vibrations  per  second,  we  can  raise  or  lower  the  pitch  of  tone 
gradually,  producing  differences  as  minute  as  the  mechanical  means  at 
our  disposal  will  allow.  There  is  no  reason  in  the  nature  of  tone  why 
we  should  select  any  one  pitch  or  rate  of  vibration  for  our  c  or  o  rather 
than  any  other.  And  even  when  a  standard  pitch  has  been  adopted 
for  practical  purposes,  minor  variations  due  to  change  of  temperature, 
mis-tuning,  etc.,  are  inevitable.  It  has  been  claimed  that  the  vibratory 
rate  of  256  should  be  taken  as  the  standard  of  'philosophic  pitch,' 
because  256  =  2^;  i.e.  if  we  imagine  a  tone  of  one  vibration  per  second, 
the  (fictional)  tone  of  two  vibrations  would  be  the  octave  of  it,  four 
vibrations  would  give  the  double  octave,  and  so  on,  so  that  the  eighth 
octave  would  give  us  256  vibrations  (50,  33  ff.).  It  is  certainly  very 
useful  to  have  a  commonly  accepted  standard  for  convenience  of 
reference.  Then  we  know  what  rate  of  vibration  is  implied  by  any 
nominal  pitch,  e.g.  A^,  without  having  to  give  it  separately.  But  the 
standard  now  perhaps  most  commonly  in  use  is  a  c^  of  264  vibrations 
per  second.  One  advantage  of  this  basis  (although  a  slight  one),  is 
that  it  is  a  multiple  of  24,  and  so  can  be  readily  used  in  connexion 
with  the  diatonic  series  of  ratios  stated  above.  I  shall  use  this  standard 
throughout  the  following  pages  unless  some  other  standard  is  specially 
indicated.  The  usually  current  nomenclature  of  octaves  may  be  looked 
upon  as  starting  from  'middle  c,'  the  c  common  to  the  baritone  and 
contralto  voice,  which  is  called  c*.  Above  that  the  octaves  are  c^,  c^,  c*, 
c^  (the  highest  note  on  the  large  concert  grand  piano),  c^,  etc. ;  below 
we  have  c^^,  C,  Cj,  C^  (A.^  is  the  lowest  note  on  the  same  instrument). 
Plain  letters  will  thus  indicate  absolute  pitch,  italicised  letters  relative 
pitch. 


CHAPTER  II 

ANALYTIC  DESCRIPTION  AND  THEORY  OF  THE  SERIES 
OF  PURE  TONES 

Having  thjus  reduced  the  usual  tonal  material  of  music  to  its  simplest 
components,  we  have  now  to  describe  this  continuous  series.  The  terms 
of  our  description,  to  be  scientifically  useful  and  explanatory,  must  be 
such  as  will  bring  tones  into  systematic  connexion  with  as  many  other 
similar  objects  as  possible. 

Being  dependent  upon  the  working  of  a  sense  organ — the  cochlea 
of  the  ear, — tones  are  classified  in  psychology  as  sensations.  We 
naturally  expect  them  to  show  great  similarity  to  the  sensations  we 
get  from  our  other  sense-organs,  such  as  the  eye  (vision),  the  tongue 
(taste),  the  skin  (touch,  temperature,  pain),  and  various  others,  such 
as  hunger  and  thirst.  The  similarity  of  all  these  to  one  another  is  certainly 
not  at  first  striking.  And  it  has  usually  been  thought  that  the  differences 
are  far  more  numerous  and  important  than  any  resemblances  there 
may  happen  to  be.  Many  men,  indeed,  judging  by  the  perennial  failure 
of  the  attempt  to  bring  our  different  senses  into  systematic  connexion, 
have  adopted  a  standpoint  of  extreme  scepticism  towards  any  such 
claim  or  expectation.  But  since  psychology  became,  some  decades 
ago,  an  experimental  science,  the  study  of  the  sensations  has  been 
pursued  most  carefully  and  exhaustively,  and  the  real  relations  of 
resemblance  and  of  structure  between  our  various  sensations  have 
gradually  grown  clearer.  Beyond  this  nothing  is  required  but  a  frank 
and  det€rmined  rejection  of  the  old  prejudice  and  a  whole  hearted 
effort  to  work  out  the  inner  similarity  of  sound  and  of  the  few  other 
senses  we  have.  The  problem  then  is  to  describe  the  tonal  series  so  as 
to  show  the  inner  connexion  not  only  between  all  the  parts  of  the 
series,  but  between  tones  and  the  sensory  objects  of  the  other  senses. 

The  first  and  most  obvious  feature  of  sounds  is  that  which  distin- 
guishes them  from  the  sensations  of  other  senses.  No  kind  of  sound  or 
group  of  sounds  is  ever  confused  with  a  sight  or  with  a  touch.  Psycho- 
logists call  this  a  difference  of  quality.  The  word  quality  is  often  used 
by  musicians  to  designate  that  difference  of  tones  of  the  same  pitch 
which  is  due  to  the  peculiar  blend  of  partials  they  contain.  It  is  better, 
however,  to  call  this  the  (pitch)  blend  of  tones.   For  practical  purposes 


6  ANALYTIC  DESCRIPTION  AND  THEORY  [ch. 

that  word  is  the  best  which  most  readily  suggests  the  thing  named, 
or  its  cause,  or  the  like.  The  word  blend  is  in  common  use  as  a  name 
for  similar  differences  in  objects  that  appeal  to  other  senses,  especially 
to  taste  and  smell,  of  which  the  latter  is  the  more  important.  The  blend 
is  here  due  to  the  mixture  of  the  components.  Similarly  the  blend  of 
a  tone  will  be  the  difference  due  to  the  admixture  of  partials,  which  a 
trained  ear  can  learn  to  pick  out  and  name,  as  a  practised  palate  will 
detect  the  components  of  a  tea  or  the  varying  flavours  of  a  wine.  This 
word  'blend'  seems  better  than  the  French  word  'timbre,'  which  does 
not  fit  into  our  language  either  in  its  native  pronunciation  or  in  ours. 

The  second  attribute  that  is  found  in  all  tones  and  in  the  sensations 
of  other  senses  is  intensity.  Both  scientific  and  popular  usage  agree 
as  to  the  meaning  of  this  term.  The  word  loudness  is  not  so  useful  for 
classification,  because  it  is  inapplicable  to  the  other  senses  and  so  does 
not  serve  to  indicate  any  variant  common  to  them  all.  One  word  of 
frequent  occurrence  must  be  carefully  avoided  in  this  connexion, 
namely,  volume.  We  think  of  loudness  as  great  volume  when  many 
instruments  sound  together  as  in  an  orchestra  and  so  make  a  very  intense 
sound  or  a  mass  of  very  many  sounds.  Having  thus  associated  many 
sounds  with  much  sound,  we  often  use  the  word  where  there  is  obviously 
only  one  sound  present,  as  when  we  speak  of  the  great  volume  of  a 
singer's  voice,  especially  of  a  contralto's  or  a  bass's.  Here  a  touch  of 
the  scientific  usage  begins  to  appear.  But  that  highly  justifiable  usage 
does  not  tolerate  any  confusion  of  volume  with  loudness. 

Volume  is  properly  used  to  distinguish  that  difference  between 
tones  of  different  pitch  that  makes  the  low  tone  great,  massive,  all- 
pervasive,  and  the  high  tone  small,  thin,  and  light.  The  other  words 
we  use  to  designate  differences  of  pitch  have  the  same  sort  of  association. 
Sharp  and  flat  are  closely  akin  to  thin  and  broad,  or  small  and  large. 
The  Latin  and  French  words  gravis,  grave,  acutus,  aigu,  bear  the  same 
implications.  In  the  eighth  Problem  on  Music  Aristotle  asked:  "Why 
does  the  low  tone  dominate  the  higher?  Is  it  because  the  low  is  the 
greater?  For  it  is  like  the  obtuse  angle,  while  the  other  resembles  the 
acute  angle."  And  the  twelfth  Problem  answers:  "Is  it  because  the 
low  tone  is  great  and  therefore  more  powerful  and  because  the  small 
is  included  in  the  great? "  (65,  17, 19;  cf.  16,  13, 19). 

Although  the  words  of  these  sentences  are  very  suggestive  to  a 
theorist  of  the  present  day,  it  is  doubtful  whether  Aristotle  meant  really 
to  ascribe  differences  of  size  to  the  tones  as  mere  sounds  or  sensations. 
His  mind  was  very  much  impressed  by  the  discovery  that  the  low  string 


n]  OF  THE  SERIES  OF  PURE  TONES  7 

gives  out  not  only  its  own  tone  but  the  higher  octave,  so  that,  as  we 
should  say  now  also,  the  low  tone  contains  'the  higher  one' — its  own 
first  higher  partial,  the  octave  of  itself.  But  Aristoxenus  refers  to  "the 
blunder  of  Lasus  and  some  of  the  school  of  Epigonus,  who  attribute 
breadth  to  tones"  (1,  167).  And  many  modern  writers  have  inclined 
more  or  less  tentatively  towards  this  idea  as  an  explicit  description  of 
tones  as  such.  We  must  now  certainly  take  the  idea  with  complete 
seriousness  and  think  of  tones  as  of  different  size  or  mass  or  bulk,  just 
as  a  visual  sensation  can  be  of  different  size  in  respect  of  its  mass  or 
area,  or  as  pain  and  hunger  can  be  large  and  massive,  or  as  pain  and 
touch  can  be  small  as  sand  or  needles.  There  is  every  reason  to  believe 
that  this  difference  of  volume  or  extent  is  dependent  upon  the  number 
of  elementary  sense-organs  of  hearing  that  are  in  action  at  the  same 
time.   But  that  is  a  question  for  physiology. 

The  only  other  property  of  pure  tones  is  what  we  commonly  call  their 
pitch.  By  pitch  tones  fall  into  a  definite  order  or  series.  This  is  not 
naturally  a  discrete  series  :  like  the  ordinal  numbers,  of  which  each 
one  is  an  individual  separated  from  the  next  by  a  unit  of  space,  into 
which  other  numbers  of  a  fractional  nature  may  be  fitted;  or  like  the 
pitches  of  the  diatonic  scale.  It  is  a  continuous  series  :  we  can  pass 
from  any  one  point  of  it  to  any  other  by  gradations  that  are  not  dis- 
tinguishable from  those  that  lie  next  to  them  on  either  side,  but  that 
are  distinguishable  from  those  that  lie  a  certain  distance  away  on  either 
side.  And  the  series  is  ordinal,  not  because  it  can  be  considered  con- 
ceptually as  a  continuous  series  of  positions,  but  because  it  appears 
so  to  us  phenomenally,  as  is  often  said,  or  merely  as  sensation.  The 
series  of  colours  of  the  spectrum,  merely  as  colours  (i.e.  apart  from  their 
position  in  the  dispersed  spectrum,  and  from  the  wave-length  they 
depend  upon)  can  be  treated  in  an  ordinal  way,  although  that  series 
is,  as  sensation,  really  a  series  of  qualities.  But  the  pitch  series  is,  as 
sensation,  itself  really  ordinal^.    It  presents  itself  to  us  as  ordinal  and 

^  Aristoxenus  wrote:  "Tension  is  the  continuous  transition  of  the  voice  from  a  lower 
position  [rdirov]  to  a  higher,"  etc.  (1, 102. 172;  14,  83f.).  The  Greek  term  is  highly  suggestive. 
Bat  it  does  not  seem  certain  that  he  meant  by  it  more  than  cessation  of  change  of 
the  voice,  a  permanence  of  one  kind  of  activity.  It  is,  of  course,  significant  that  in  this 
case  we  naturally  incline  towards  the  term  'place'  or  to  the  idea  of  the  voice's  'moving.' 
We  do  not  incline  to  say  that  the  weather  '  moves,'  or  the  colours  of  leaves  '  move '  in  the 
autumn,  when  we  mean  only  that  they  change.  No  doubt  Aristoxenus  used  the  terms 
'place'  and  'motion'  at  the  suggestion  of  the  ordinal  and  motional  aspects  of  tone.  Never- 
theless his  concepts  of  position  and  motion  of  the  voice  probably  did  not  include  more 
than  what  he  might  have  attributed  to  a  thing  that  only  changes,  i.e.  progress  of  change 
and  arrest  of  change   Cf.  80,  asit. 


8  ANALYTIC  DESCRIPTION  AND  THEORY  [ch. 

calls  for  ordinal  names,  whether  we  know  anything  about  the  wave- 
lengths that  cause  it  or  not. 

The  attributes  of  tones  thus  far  enumerated  are:  quality,  intensity, 
volume  and  pitch.  The  relations  between  these  four  are  an  important 
problem.  It  has  been  suggested  that  intensity  is  a  sort  of  density  of 
sensation,  as  it  were.  Just  as  a  gas  may  fill  a  certain  volume  and  yet 
be  very  thinly  scattered  throughout  it,  so  it  is  thought  a  sensation 
may  be  of  one  and  the  same  quality,  and  volume,  and  pitch,  and  yet 
be  more  or  less  dense — or  intense.  Suggestive  arguments  in  favour  of 
this  view  have  been  advanced,  but  they  do  not  yet  seem  sufficient  for 
their  purpose. 

The  relation  between  pitch  and  volume  is  much  clearer.  When 
tones  are  compared  with  noises,  a  marked  difference  is  apparent.  Tones, 
as  everyone  feels  and  knows,  are  smooth  and  regular,  noises  are  rough 
and  irregular.  Tones  may  also  be  said  to  be  balanced  and  symmetrical, 
while  noises  are  chaotic  and  disorderly.  These  descriptions  obviously 
refer  to  the  volume  of  tones,  not  to  their  pitches.  A  pitch  has  only 
a  definite  position  or  place;  it  is  not  smooth  or  balanced.  But  pitch 
gives  tone  a  position  as  a  whole;  it  is  by  means  of  pitch  that  tones  are 
brought  into  a  definite  and  accurate  series,  and  their  volumes  along 
with  them.  The  question  then  arises  :  what  position  has  pitch  itself 
in  the  tone's  volume? 

This  is  not  an  absurd  question,  but  a  very  natural  one.  For  if  tones 
have  an  aspect  of  volume  and  can  be  arranged  in  a  very  definite  and 
single  series  by  means  of  pitch — a  property  that  is  distinguishable 
from  volume;  and  if  pitch  is  not  only  thus  really  ordinal,  but  is  also 
felt  as  ordinal  or  appears  to  us  so  as  a  property  of  sensation;  it  is 
perfectly  natural  to  suppose  that  what  is  thus  ordinal  is  a  part  of  the 
tone's  volume  and  to  ask  in  consequence — which  part  of  the  volume 
constitutes  the  pitch? 

Of  course,  one's  habits  of  thought  may  oppose  this  line  of  inquiry. 
One  of  the  greatest  obstacles  to  the  advance  of  knowledge  is  the 
opposition  our  minds  offer  by  the  mere  force  of  unfamiliarity  to  the 
application  of  old  and  simple  notions  to  common  objects  to  which  they 
have  not  hitherto  been  applied.  The  mind  seems  to  refuse  to  establish 
the  desired  connexion.  All  sorts  of  excuses  and  objections  are  offered 
to  the  new  invitation.  "Metaphors  are  so  misleading."  But  it  is  not 
a  case  of  metaphors  now.  Pitch  is  no  mere  analogy;  the  ordinal  status 
and  arrangement  of  tones  is  one  of  the  bed-rock  facts  of  music.    And 


n]  OF  THE  SERIES  OF  PURE  TONES  9 

'volume'  is  no  more  a  mere  simile  than  is  interval  or  concord  or  discord. 
It  is  as  much  there  as  any  fact  could  possibly  be.  Psychologists  admit 
it  more  and  more  frequently,  and  it  is  only  a  matter  of  time  till  everyone 
who  considers  the  subject  will  agree  with  them.  Nor  is  it  'mystical'  to 
suggest  that  pitch  has  a  position  in  volume.  A  line  of  well-founded  and 
logical  thought  is  only  mystical  to  those  who  do  not  take  the  trouble 
to  foUow  it  carefully.  A  mystic  is  one  who  claims  to  have  special  insight 
or  experience  which  he  has  discovered  by  accident  or  providential 
good-will  and  which  he  is  powerless  to  reveal  to  others  either  because 
it  defies  all  description  or  because,  not  knowing  how  he  himself  attained 
it,  he  is  unable  to  lead  thither  all  who  would  share  it  with  him.  But 
there  is  nothing  mystical  about  pitch  or  tonal  volume;  nor  are  the 
ordinary  logical  processes  of  inference  held  to  be  the  special  privilege 
of  a  few  minds. 

We  may  therefore  consider  our  question  clear  and  reasonable.  And 
the  most  likely  answer  follows  naturally  from  the  apparent  balance 
and  symmetry  of  tones  as  compared  with  noises.  We  may  assume  that 
pitch  holds  a  central  position  in  volume.  And,  as  pitch  is  ordinal,  while 
volume  suggests  a  volume  of  parts  or  particles,  we  may  go  on  to  assume 
that  pitch  is  constituted  by  a  specially  prominent  or  noticeable  part 
of  the  volume  of  sound  that  makes  up  a  tone.  We  certainly  do  not 
hear  tone  as  a  group  of  distinguishable  particles  like  a  handful  of  sand. 
We  hear  it  as  a  continuously  smooth  closed  volume.  But  nevertheless 
a  part  of  this  whole  volume  might  well  be  more  noticeable  than  the 
rest,  just  as  a  part  of  a  variably  'toned'  visual  surface  may  be  most 
deeply  coloured — red,  for  example, — although  we  could  not  pick  out 
and  isolate  any  part  of  it  that  would  be  all,  and  nothing  less  or  more 
than  all,  the  reddest  part.  Yet  no  one  doubts  that  a  visual  surface 
consists  of  a  mass  of  minimal  particles  of  colour  surface,  grouped  into 
a  continuous  whole.  These  particles  are  presumably  the  minimal  areas 
of  colours  given  by  single  recipient  visual  organs — the  cones  (and  rods) 
of  the  retina.  There  are  also  in  the  ear  elementary  receptors  of  sound; 
and  these  presumably  afford  us  the  minimal  particles  of  sound  that 
make  up  tone. 

Now  all  (pure)  tones  are  the  same  in  symmetry  and  balance  and 
smoothness.  So  we  may  consider  this  central  position  and  predominance 
of  pitch  to  be  characteristic  of  pure  tone  as  against  all  grades  of  noise, 
which  are  relatively  rough  and  unbalanced,  vague  or  indefinite  in  pitch 
or  marked  by  many  prominent  points  of  pitch.  Tones  differ  from  one 
another  in  size  of  volume  and  in  the  ordinal  position  of  their  pitches 


10 


ANALYTIC  DESCRIPTION  AND  THEORY 


[CH. 


relatively  to  one  another.  The  pitch  of  a  higher  tone  lies  a  little  to  one 
side  of  the  pitch  of  a  tone  just  lower  in  pitch;  and  the  pitches  of  all 
tones  together  form  a  single  linear  series,  having  the  tone  of  greatest 
volume  at  one  end  and  the  tone  of  least  volume  at  the  other.  If  we 
were  to  project  the  volumes  and  the  pitches  of  all  the  tones  of  the  series 
against  one  another  in  our  thought,  we  should  obtain  a  scheme  of  the 
following  kind: 

High  Tones  _^ 


r 


1P- 


Low  Tones 


P 

Kg.  1 


P' 


^h 


If  we  placed  all  the  pitch-points  on  a  perpendicular  line  above  one 
another,  we  should  indeed  represent  the  decrease  of  volume  (as  we  go 
up)  properly  by  the  decrease  in  the  breadth  of  the  line  used,  while  the 
symmetry  and  balance  of  tone  would  be  indicated  by  the  central  position 
of  the  P  point  in  the  volume  line  ( F?  =  '  lower '  end  of  volume, 
Vh  =  'higher'  end  of  volume).  But  we  should  not  have  given  any 
representation  of  the  fact  that  the  pitch  of  a  tone  higher  than  another 
lies  on  one  side  of  the  pitch  of  the  latter  in  an  ordinal  series.  This  series 
is  quite  properly  indicated  in  our  figure. 


We  have  as  yet  no  proof  for  the  assumption  that  the  Vh  ends  of 
all  the  volumes  should  lie  perpendicularly  above  one  another,  or — 
whether  in  mere  projection  on  the  base  line  of  the  figure  (which  may 
be  supposed  to  represent  the  greatest  possible  volume  or  the  lowest 
possible  tone),  or  in  reality — should  be  the  same  point.  It  is  conceivable 
that  the  pyramid  of  tones  should  be  acute  or  obtuse  angled  rather 
than  right-angled.  But  these  alternatives  are  far  from  likely  for  various 
reasons  of  which  the  most  important  will  be  set  forth  immediately. 


II]  OF  THE  SERIES  OF  PURE  TONES  11 

It  would  follow  from  our  scheme  that  if  the  projected  series  of  pitches 
is  in  any  way  real,  that  ought  to  appear  in  an  unmistakable  manner 
when  tones  of  different  pitches  are  given  simultaneously.  And  this  is 
the  case.  Simultaneous  tones  seem  to  be  mixed  together  or  to  fuse 
with  one  another  or  to  intermingle.  They  never  appear  to  be  entirely 
apart  from  one  another  as  two  patches  of  colour  often  do,  when  they 
are  separated  by  a  patch  of  a  third  colour  or  when  they  just  bound 
one  another.  Each  patch  of  colour  is  seen  as  well  when  both  are  given 
together  as  when  they  appear  singly  and  successively.  Not  so  two 
tones;  they  always  appear  to  be  in  each  other's  way,  to  crowd  upon 
or  to  overlap  one  another.  This  holds  even  for  the  greatest  extremes, 
when  the  highest  and  lowest  tones  are  given  together.  And  yet  at  the 
same  time  the  two  tones  by  no  means  completely  lose  their  individuality 
or  become  indistinguishable  in  this  'mixture,'  as  two  colours  do  when 
they  are  mixed  by  being  cast  upon  the  same  surface  or  by  the  rotation 
of  them  on  a  disc.  Blending  of  colours  gives  a  new  colour  in  which  the 
components  are  essentially  indistinguishable  by  any  one  who  did  not 
see  them  before  their  mixture.  But  musical  folks  can  detect  the  com- 
ponent tones  of  a  chord  with  ease  and  certainty.  In  spite  of  the  over- 
lapping or  interpenetration  of  tones  they  are  more  or  less  readily 
distinguishable.  Their  pitches  generally  strike  us  as  being  the  same  in 
mixture  as  in  isolation. 

If,  as  we  have  supposed,  the  series  of  pitches  represents  a  series 
of  real  particles  of  sound,  a  ready  explanation  of  this  peculiar  inter- 
mingling of  tones  is  at  hand.  Then  the  sounds  that  form  the  highest 
musical  tone  will  form  the  last  (or  highest)  part  of  the  volume  of  all 
simultaneous  lower  tones.  Being  the  same  sounds,  the  two  common 
parts  will  overlap  or  intermingle;  but,  as  the  higher  tone  adds  its 
intensity  to  the  high  part  of  the  low  tone,  and  especially  the  predominant 
intensity  of  its  pitch,  the  high  tone  will  stand  forth,  and  be  detectable 
in  the  lower  tone  in  spite  of  the  overlapping  of  the  two.  And  the 
rectangular  shape  of  the  scheme  we  have  figured  is  justified.  The 
possibility  of  an  obtuse-angled  figure  is  then  excluded;  for  that  would 
imply  that  the  volume  of  the  highest  tone  lay  beyond  the  volume  of 
the  lowest  tone,  so  that  a  medium  tone  would  not  mix  with,  or  obscure, 
a  very  high  tone  in  the  slightest  degree.  And  the  probability  of  an 
acute-angled  form  being  the  true  scheme  is  equally  small;  for  in  that 
case  the  movement  of  pitch  that  accompanies  the  continuous  decrease 
of  volume  would  proceed,  as  it  actually  does,  steadily  in  one  direction, 
but  only  up  to  a  certain  point,  after  which  the  direction  would  be 


12  ANALYTIC  DESCRIPTION  AND  THEORY  [ch. 

reversed.  Of  this  there  is  no  actual  trace  in  hearing.  We  may  therefore 
for  the  present  safely  follow  the  scheme  depicted  in  the  figure.  We  shall 
obtain  further  evidence  on  this  point  when  we  come  to  the  study  of 
the  grades  of  fusion. 

We  conclude,  therefore,  that  tone  is  a  mass  or  volume  of  minute 
(hypothetical)  particles  of  sound  sensation,  of  which  those  at  its  centre 
are  the  most  intense,  while  the  others  grade  themselves  on  either  side 
in  the  whole  volume  so  that  the  mass  appears  regular  and  balanced. 
Since  in  a  pure  tone  no  other  predominating  points  or  pitches  appear, 
we  must  suppose  that  the  intensity  of  its  constituent  particles  decreases 
gradually  and  regularly  from  the  central  pitch  towards  the  limits  of 
the  volume^.  In  this  respect  every  tone  is  appreciably  the  same,  no 
matter  what  its  pitch  may  be.  The  particles  that  constitute  volume 
must  be  called  hypothetical,  because  the  volume  does  not  appear  to 
be  a  group  of  distinguishable  parts.  We  do  not  separately  experience 
the  minimal  particle  except  possibly  in  the  case  of  the  tone  or  sound 
of  the  smallest  possible  volume,  i.e.  the  highest  audible  sound.  That 
may  be  approximately  the  minimal  particle.  Between  this  highest 
particle  (or  pitch)  and  the  pitch  of  any  tone  there  lies  the  whole  series 
of  pitches  that  leads  from  the  latter  to  the  former.  And  we  have  reason 
to  infer  that  half  of  the  volume  of  any  tone  is  made  up  of  the  pitch 
particles  that  appear  in  the  series  of  tones  progressively  higher  than 
itself  up  to  the  highest  tone.  So  the  lowest  audible  tone  would  consist 
in  its  one  ('upper')  half  of  the  whole  series  of  audible  pitch -points. 
The  other  (or  'lower')  half  of  its  volume  is  not  used  for  the  formation 
of  the  pitches  of  other  tones.  That  is,  of  course,  no  argument  against 
its  existence. 

Every  tone  must,  therefore,  contain  within  it  the  volumes  of  any 
higher  tone.  And  all  tones  are  constituted  from  a  single  series  of  sound 
particles,  of  which  they  incorporate  a  series  always  beginning  at  the 
common  upper  end  and  stretching  downwards  as  far  as  the  size  of  each 
tone's  volume  requires.  These  and  other  such  relations  can  easily  be 
read  from  the  figure  already  given. 

*  Wm.  Gardiner  in  his  popular  compendium  of  musical  topics.  The  Music  of  Nature 
(13,  188)  gives  a  curious  diagram  entitled:  "The  wind  instruments — the  shape  and  order 
of  their  tones  from  the  lowest  to  the  highest,"  which  shows  a  column  of  twenty-one  oval 
figures,  coloured  differently  for  the  different  instruments,  with  a  small  dot  at  the  centre 
of  each.  These  ovals  grow  gradually  smaller  towards  the  top  of  the  column.  They  might 
all  be  inscribed  in  an  angle  whose  sides  were  about  five  centimetres  long  and  two-thirds 
of  a  centimetre  apart  at  the  ends.  Gardiner  did  not  in  any  way  elucidate  in  words  the 
reasons  that  led  to  this  diagram. 


II]  OF  THE  SERIES  OF  PURE  TONES  13 

The  series  of  pitches  may  be  said  to  define  clearly  one-half  of  one 
of  the  dimensions  of  tonal  volume.  We  say  *one,'  because  we  do  not 
feel  tone  to  be  a  mere  line  or  length,  but  a  volume;  something  areal 
or  massive  or  round,  as  it  were.  Of  course,  in  so  far  as  we  think  of  tones 
from  the  pitch  point  of  view  they  fall  into  a  perfectly  definite  ordinal 
or  linear  series  of  no  thickness  at  all,  so  to  speak.  Yet  when  we  look 
upon  tone  naively  as  a  whole,  it  is  as  mass  or  volume  that  it  appears 
before  us.  This  implies  that  it  has  at  least  one  other  dimension,  different 
from  that  indicated  by  pitches.  The  musical  aspects  of  tone  give  us 
no  means  of  demonstrating  or  of  defining  this  direction.  For  these 
aspects  are  concerned  only  with  the  variations  of  tone  in  pitch  and 
volume,  i.e.  only  with  longitudinal  variations.  And  no  transverse 
variation  accompanies  these.  We  must  turn  to  quite  a  different  function 
of  hearing,  one  that  is  dependent  not  on  either  ear  separately  or  equally, 
but  on  both  ears  integratively  or  in  a  combined  purpose.  This  function 
enables  us  to  localise  sounds  towards  the  right  or  the  left  ear  and  in  an 
imperfect  way  round  the  head  in  space.  A  careful  study  of  binaural 
hearing  leads  to  the  conclusion  that  every  tone  has  breadth  as  well 
as  length.  This  breadth  is  also  marked  out  into  a  series  by  the  variations 
of  binaural  localisations  or  'local  signs';  and,  unlike  the  pitch  series, 
this  one  is  traversable  as  far  towards  the  one  end  ('opposite  the  one  ear') 
as  towards  the  other  ('opposite  the  other  ear'). 

We  have  no  means  of  comparing  or  of  measuring  the  extents  of  the 
two  series  with  one  another,  such  as  superposition  or  the  like.  But  we 
do  not  feel  this  ignorance  or  disability  as  a  difl&culty  or  mystery;  for 
we  simply  do  not  think,  we  have  no  inclination  to  think,  of  these  series 
in  relation  to  one  another.  On  the  contrary,  the  one  is  the  musical 
aspect  of  tone,  the  other  is  the  basis  of  its  spatial  aspect.  Music  is  the 
same  for  a  person  of  normal  hearing  whether  it  is  played  to  the  right 
or  to  the  left  of  him.  And  the  tonal  nature  of  a  warning  signal  is  largely 
insignificant,  if  we  but  gauge  the  position  of  its  source  aright.  These 
two  interests  of  sound  appeal  to  very  different  practical  functions  in 
spite  of  the  close  relation  of  their  ultimate  bases  to  one  another.  Besides, 
the  total  transverse  breadth  of  tone  probably  hardly  ever  varies,  unless 
in  passing  from  purely  uniaural  to  binaural  hearing.  It  is  only  the 
point  of  emphasis  in  the  total  breadth  that  varies  and  so  forms  a  basis 
for  localisation  towards  the  one  ear  or  the  other.  The  longitudinal  or 
musical  aspects  of  tone  would,  therefore,  be  unaffected  in  any  way. 
And  in  case  there  might  be  any  diversion  of  interest,  we  neither  en- 
courage our  musicians  to  rove  around  us  while  performing,  nor  do  we 


14       DESCRIPTION  AND  THEORY  OF  PURE  TONES      [ch.  n 

practise  an  oscillation  between  hearing  in  the  usual  way  and  hearing 
with  one  ear  only.  But  even  though  the  breadth  of  tone  is  not,  and 
from  the  nature  of  the  case  cannot  be,  a  musical  variant,  it  is  still  there 
all  the  time  in  musical  sound,  and  doubtless  makes  it  what  it  is — a 
volume.  (For  proof  of  this  transverse  aspect  of  sound,  see  77,  Chap,  ix.) 

We  may  sum  up  our  conclusions  by  saying  that  pure  tone  is  a  volume 
of  sound  in  which  a  minute  part  or  point  is  most  intense,  while  the 
rest  is  graded  smoothly  and  symmetrically  around  this — probably 
central — part,  the  pitch  of  the  tone.  By  means  of  pitch,  tones  can  be 
arranged  in  a  series  of  diminishing  volumes.  But  no  two  simultaneous 
tones  fail  to  fuse  or  blend  with  one  another  or  to  be  heard  through 
one  another.  We  therefore  infer  that  the  volume  of  any  tone  includes 
not  only  the  pitch,  but  the  whole  volume  of  every  higher  tone;  so  that 
all  tones  may  be  reduced  to  a  single  series  of  hypothetical  particles 
of  sound,  one  half  of  which  series  we  actually  hear  as  the  pitches  of 
tones.  Apart  from  these  pitch  points  the  nearest  approach  to  the 
hypothetical  particle  of  sound  that  is  ever  separately  experienced  by 
us  is  the  minute  volume  of  the  highest  audible  tone  or  sound.  The 
other  dimension  of  tonal  volume  appears  in  the  '  local  signs '  of  binaural 
hearing.  But  there  is  no  method  whereby  we  might  compare  the  size 
or  length  of  this  series  with  the  length  of  the  pitch  series,  so  as  to  say 
what  the  exact  shape  of  tonal  volume  is.  We  must  be  content  with  the 
close  correspondence  shown  between  the  feeling  of  tonal  volume  and 
what  we  have  proved  it  to  be.  The  probable  shape  of  tone  is  like  a 
visual  parallelogram  that  is  longer  than  it  is  broad.  It  varies  greatly 
in  its  length  but  probably  not,  or  hardly,  at  all  in  its  breadth.  This 
relation  between  breadth  and  length  constitutes  the  typical  and  constant 
form  of  tonal  volume.   It  may  be  illustrated  diagrammatically  thus: 

V 


\1  L 


Ve  h 


If  ^_______^  L 


Vo  h 


Fig.  2.  Diagram  of  two  tones  a  tenth  apart,  e.g.  c^  and  e*,  showing  the  constant  breadth  (6) 
and  the  variable  lengths  {Lfi-L  and  IP-L)  of  the  volumes  of  the  tones  ( Vo  and  Ve), 
and  the  intensive  differences  within  the  volume  (2/),  the  pitch  (j?)  being  the  pre- 
dominant point  of  the  whole. 


CHAPTER  III 

DEGREES  AND  THEORY  OF  CONSONANCE  AND 
DISSONANCE  (FUSION) 

In  a  general  way  the  various  grades  of  consonance  between  two  tones 
sounded  simultaneously  have  been  long  established  and  are  disputed 
by  no  one.  The  ancient  Greeks  recognised  three  grades  of  consonance 
or  symphony — the  octave,  the  fifth,  and  the  fourth.  To  these,  in  a 
certain  sense^,  the  third  (but  not  the  sixth)  was  added  by  Garudentius 
about  the  end  of  the  third,  or  the  beginning  of  the  fourth,  century  a.d. 
(cf.  66,  72).  Modern  musical  theory  grades  the  consonances  as  perfect — 
octave,  fifth,  and  fourth,  and  imperfect — the  major  and  minor  thirds 
and  sixths.  All  the  other  intervals  smaller  than  the  octave  are  dis- 
sonant— the  sevenths,  the  tritone  (diminished  fifth  or  augmented 
fourth)  and  the  seconds.  The  increase  of  any  interval  by  an  octave 
is  not  held  to  make  any  change  in  its  consonance  or  dissonance  (except, 
in  order  to  satisfy  the  needs  of  certain  theories,  in  certain  chords). 

Of  the  dissonances,  however,  the  minor  seventh,  as  also  sometimes 
the  tritone,  is  commonly  considered  to  be  almost  consonant.  Frequent 
reference  is  made  (especially  by  those  who  look  to  the  series  of  partials 
for  the  causes  or  origins  of  chords  and  scales)  to  the  approximate 
equality  between  the  diminished  seventh  and  the  'natural'  seventh 
formed  between  the  fourth  and  the  seventh  harmonics  of  any  funda- 
mental (in  the  case  of  the  tritone  the  fifth  and  seventh  harmonics). 
The  difference  is  only  a  sixty-fourth  part — in  terms  of  the  ratios  of 
vibration  of  the  two  partials.  It  has  often  been  claimed  that  the  chord 
cegb\f,  taken  exactly  in  the  ratios  4,  5,  6,  7,  is  really  a  concord  (cf. 
66,  71).  Those  who  look  for  some  fundamental  development,  say  of  the 
ear  itself,  underlying  the  progress  of  musical  art,  might  claim  this 
tendency  as  another  step  forwards  beyond  the  one  first  recorded  by 
Gaudentius.  Various  theorists  have  even  believed  that  all  the  grades 
of  consonance  have  their  ground  in  habit,  racial  if  not  individual.  For 
the  partial  that  occurs  oftenest  and  loudest  with  any  fundamental  is 
probably  its  octave,  the  next  is  the  fifth  above,  and  so  on  in  the  series 
of  partials.    What  we  hear  oftenest  together,  the  mind  (individually)  or 

*  Which  we  shall  have  occasion  to  examine  more  closely  later  on. 


16  DEGREES  AND  THEORY  [oh. 

the  ear  (racially)  comes  to  hear  as  one  or  at  least  as  an  agreeable  combina- 
tion— because  the  ear  or  the  mind  has  got  accustomed  to  hearing 
them  together.  Thus  in  its  progress  the  art  of  music  is  gradually 
climbing  the  ladder  of  partials.  We  have  already  accommodated 
1,  2,  3,  4,  5,  and  6  in  all  possible  groupings  and  we  are  now  absorbing  7. 
In  some  future  we  shall  bring  even  9,  and  11,  and  13  and  others  that 
we  now  treat  as  dissonances,  over  the  border  as  consonances^.  One 
writer  has  even  tried  to  create  an  experimental  basis  for  this  theory 
by  giving  persons  much  practice  in  placid  attention  to  dissonances. 

A  number  of  experimental  investigations  have  been  made  with  the 
purpose  of  grading  the  diatonic  intervals  more  accurately  than  the 
usage  of  music  indicates.  The  general  result  of  these  yields  the  series  : 
octave,  fifth,  fourth,  major  third,  minor  third,  major  sixth,  minor 
sixth,  tritone,  major  second,  minor  seventh,  minor  second,  major 
seventh;  or  in  a  useful  symbolism,  which  will  be  maintained  in  the 
following  pages,  0,  5,  4,  III,  3,  VI,  6,  T,  II,  7,  2,  VII.  This  series 
represents  comparatively  to  what  degree  the  two  tones  seem  to  fuse 
with  one  another  to  form  a  unitary  whole.  0  is  more  fused  than  5,  5  than 
4,  and  so  on. 

In  other  experiments  an  attempt  has  been  made  to  obtain  figures 
representative  of  the  degree  of  difference  between  the  grades  of  fusion. 
Highly  trained  musical  ears  might  attempt  to  indicate  these  quantities 
by  direct  estimation.  But  their  judgments  are  quite  open  to  the  influence 
of  their  knowledge  of  musical  practice  or  of  any  other  thoughts  or 
theories  they  may  have.  For  they  recognise  the  interval  as  soon  as  it 
is  sounded  and  so  can  think  of  it  whatever  occurs  to  them.  And  the 
relative  quantity  of  fusion  is  not,  as  it  were,  marked  on  the  face  of 
intervals.  More  or  less  unmusical  minds,  however,  are  often  quite 
unable  to  distinguish  the  two  pitches  of  an  interval,  or  to  recognise 
one  of  them  or  the  interval  as  a  whole,  etc.  Consequently  they  think 
nothing  about  them  by  way  of  memory,  so  that  they  are  sure  to  be 
freer  from  suggestive  influences  than  musical  minds.    This  advantage 

1  Cf.  A.  E.  Hull,  22,  265:  "The  standard  of  aesthetics  varies  from  age  to  age.  A  com- 
bination of  notes  which  one  generation  accepts  only  on  sufferance  will  be  received  by  a 
later  generation  with  equanimity  or  even  deUght:  Monteverde's  Sevenths,  Wagner's 
Ninths,  Gounod's  Thirteenths,  Debussy's  Twelfths,  and  so  on."  On  page  115  Hull  gives 
a  scheme  of  development  which  places  the  first  two  partials  as  primeval,  the  second  two 
as  mediaeval,  5,  6,  7,  8  and  9  as  of  the  18th  and  19th  centuries,  10  to  16  in  the  form  of 
e,/JJ,  g,  at?,  hv,  h  and  c  as  "whole-tone  scale,  Debussy,  Scriabin."  "Undoubtedly  Scriabin's 
exploitation  of  the  higher  harmonics  wiU  lead  to  wonderful  developments,  which  are 
even  already  in  evidence"  (p.  271).  But  he  says  elsewhere  (p.  265),  "Many  passages 
in  Scriabin's  work  seem  ugly  to  us,  some  almos  repulsively  so."  Cf.  24, 


in]  OF  CONSONANCE  AND  DISSONANCE  17 

weighs  against  their  want  of  practice  and  skill,  which  can  even  be  turned 
to  special  account.  For  the  unmusical  mind's  failure  to  detect  the 
pitches  in  intervals  may  be  used  to  obtain  an  index  of  their  grades  of 
fusion.  The  oftener  one  fails  to  detect  that  two  sounds  have  been 
presented,  the  more  unitary  and  fused  we  may  suppose  the  whole  sound 
mass  to  have  been;  the  oftener  the  listener  feels  that  two  sounds  have 
been  given,  the  less  unitary  and  fused  must  the  combined  sound  have 
appeared  to  him.  One  series  of  experiments  in  this  manner  gave  80  % 
of  answers  asserting  the  presence  of  one  tone  where  there  had  really 
been  an  octave ;  50  %  where  there  had  been  a  fifth ;  35  %  for  the  fourth ; 
30  %  for  the  minor  third;  27  %  for  the  major  third;  23  %  for  the  tritone. 
These  and  other  results  have  shown  that  there  is  a  marked  difference 
between  the  octave  and  the  fifth,  and  between  the  fifth  and  all  the 
others;  but  that  amongst  these  last  there  are  at  the  most  only  slight 
differences  of  quantity.  If  enough  tests  are  made  carefully  the  usual 
grading  will  emerge  on  the  average.  But  just  as  in  the  tests  for  grading 
without  respect  to  quantity,  the  single  tests  here  show  frequent  reversal 
of  the  results  that  appear  on  the  average.  This,  of  course,  only  emphasises 
the  approximate  equality  of  the  lower  grades  of  fusion. 

Those  who  are  not  familiar  with  the  notion  of  fusion  must  be  careful 
to  avoid  misapprehension.  The  results  do  not  suggest  that  intervals 
such  as  the  fourth  and  the  thirds  are  hardly  distinguishable.  Fusion 
does  not  apply  to  the  peculiar  aspect  of  a  pair  of  tones  that  we  call 
their  interval;  but  to  the  whole  mass  of  sound  formed  by  the  two 
tones  in  so  far  as  they  merge  into,  or  blend  with,  one  another,  or  in  so 
far  as  they  appear  to  be  one  to  a  person  who  does  not  recognise  them 
as  one  interval  or  another,  and  yet,  of  course,  hears  them  as  a  whole. 
Nor  do  these  grades  of  fusion  imply  that  the  musician  does  not  know 
and  hear  the  fourth  as  a  greater  consonance  than  the  third,  or  the  third 
than  the  second.  The  musician  is  highly  practised;  he  has  learnt  by 
long  experience,  from  tradition,  and  by  harmonic  usage  to  classify  the 
pairs  of  tones  as  distinct  grades  of  consonances  in  spite  of  the  slight 
differences  that  may  distinguish  them,  so  that  these  differences  perhaps 
seem  greater  and  more  decisive  to  him  than  they  really  are.  This 
distinctness  of  grading  is  very  much  increased  by  the  sharp  differences 
there  are  between  pairs  of  tones  as  intervals,  whereby  they  are  at  once 
recognisable  and  distinguishable.  For  whatever  is  musically  associated 
with  a  certain  pair  of  tones  can  be  attached  to  their  distinctive  nature 
as  intervals  and  so  can  be  unfailingly  recalled;  whereas  if  it  had  to  be 
recalled  by  their  indistinctive  nature  as  fusions  alone,  the  slightness  of 

w.  r.  M.  2 


18  DEGREES  AND  THEORY  [ch. 

the  difference  between  the  lower  grades  of  fusion  would  make  recall 
very  uncertain  and  liable  to  confusion,  so  that  in  turn  the  want  of 
distinction  amongst  the  lower  grades  would  be  thrown  into  prominence. 
In  observing  fusion  the  musician  has  the  special  difficulty  of  abstracting 
from  his  highly  trained  knowledge  and  from  the  intervallic  aspect  of 
tonal  pairs.  The  Greeks,  of  course,  could  easily  distinguish  thirds  from 
seconds  as  intervals,  but  it  took  them  long  to  compare  the  members 
of  their  class  of  discordant  intervals  with  one  another  so  carefully  as 
to  see  that  the  third  stood  very  high  in  the  class  and  had  so  much  fusion 
in  it  that  they  could  place  it  next  to  the  fourth  and  even  class  it  in 
a  certain  sense  as  a  consonance.  Advance  in  observation  and  analytic 
abstraction  seems  to  account  for  their  change  of  classification  more 
easily  than  the  hypothesis  of  development  does.  Improvements  in  the 
method  of  tuning  may  also  have  been  of  influence.  But  there  is,  as  we 
shall  see,  no  doubt  that  the  differences  between  all  the  intervals  with 
which  we  are  so  familiar  nowadays  has  only  been  fully  displayed  by 
the  functions  they  have  acquired  in  our  highly  developed  music. 

The  explanation  of  fusion  in  the  first  place  follows  closely  the 
suggestions  given  by  the  description  of  its  highest  grades.  Fusion  is 
degree  of  resemblance  to  the  unity  or  balance  of  a  single  tone.  Two 
tones  that  fuse,  blend  with  one  another  so  as  to  appear  more  or  less 
evenly  intermingled,  whether  the  hearer  fails  to  recognise  their  difference 
or  whether  his  natural  aptitude  and  his  practice  enable  him  to  recognise 
them  at  once.  The  fusion  is  not  altered  by  the  ability  to  recognise 
the  constituent  tones  in  spite  of  their  fusion.  To  the  talented  musician 
who  practically  never  fails  to  notice  both  its  tones,  an  octave  is  still 
a  high  grade  blend.  Nor  does  his  ear  contrive  to  isolate  these  two 
tones  from  one  another  so  that  they  shall  appear  to  him  to  be  as  separate 
as  they  are  when  successive. 

Our  previous  study  of  tone  showed  that  the  whole  series  of  tones 
from  highest  to  lowest  probably  consists  of  one  total  series  of  (hypo- 
thetical) particles  of  sound,  of  which  a  number  always  beginning  at 
the  same  ('high')  end  of  the  series  enters  into  any  tone  sufficient  to 
make  up  its  volume.  The  highest  audible  tone  requires  approximately 
only  the  single  ('highest')  terminal  particle,  the  lowest  the  whole 
series.  Now  when  two  tones  are  given  together,  the  volume  of  the 
lower  must  include  the  volume  of  the  higher  and  apart  from  some 
special  marks  the  two  will  not  be  very  easily  distinguishable.  The 
coincidence  of  the  two  volumes  would  probably  of  itself  make  the  upper 


m]  OF  CONSONANCE  AND  DISSONANCE  19 

part  of  the  total  volume  more  intense.  And  it  is  even  conceivable  that 
the  upper  volume  might  thereby  be  recognisable  as  such,  so  that  the 
listener  could  say  :  in  this  total  sound  there  is  besides  the  total  extent 
of  sound,  a  sound  of  a  certain  smaller  extent.  And  it  may  be  supposed 
that  these  sounds  would  appear  even  and  smooth  (or  specifically  tonal) 
in  so  far  as  other  and  irregular  changes  of  intensity  are  absent  in  the 
two  volumes. 

From  this  point  onwards  the  attempt  might  be  made  to  construe 
the  whole  nature  of  tones  and  their  combinations  without  the  use  of 
pitch  in  the  sense,  above  expounded,  of  a  central  point  of  prominence 
in  the  tone's  volume.  I  shall  not  argue  the  attempt  out  in  detail.  It 
will  suffice  to  acknowledge  its  logical  possibility  here.  The  use  made 
of  the  notion  of  pitch  in  the  following  pages  will  of  itself  exclude  the 
real  possibility  of  doing  without  it. 

Pitch,  as  above  shown,  probably  occupies  a  central  position  in  tone 
and  makes  it  the  balanced  symmetrical  system  it  is.  If  this  is  so,  the 
overlapping  of  volumes  will  yield  a  special  case  when  the  higher  volume 
lies  exactly  between  the  pitch  of  the  lower  volume  and  their  common 
higher  terminal  (cf.  fig.  7,  p.  199,  c  and  c^;  or  fig.  8,  p.  200).  The  whole 
volume  thus  constituted  will  differ  from  the  isolated  volume  of  the 
lower  tone  only  in  the  extra  intensity  of  the  upper  half  and  in  the  point 
of  predominance  that  is  the  pitch  of  the  higher  tone.  In  so  far  as  this 
pitch  is  detectable,  the  upper  tone  would  be  as  precisely  definable  as  in 
isolation.  And  in  so  far  as  the  lower  end  of  the  upper  tone  can  be  felt 
to  coincide  exactly  vdth  the  pitch  of  the  lower  tone,  or  at  least  to  deviate 
from  it  when  it  does  not  so  coincide,  this  particular  case  of  simultaneous 
tones  would  be  very  precisely  definable.  And  the  whole  sound  would 
approximate  more  nearly  to  the  nature  of  a  single  pure  tone  than  would 
any  other  form  of  coincidence  of  volumes  and  pitches.  These  things 
justify  completely  the  identification  of  the  octave  fusion  with  this 
special  case  of  overlapping. 

A  second  special  case  would  be  given  when  the  two  special  defining 
points  of  the  higher  volume  lay  equally  far  away  on  either  side  from 
the  predominating  point  of  the  lower  tone  (cf.  fig.  7,  p.  199,  c  and  g). 
This  case  would  give  a  lesser  approximation  to  the  balance  of  the  pure 
tone  than  does  the  octave.  For  in  it  two  new  points  or  breaks  of  the 
smooth  continuity  of  the  lower  volume  have  been  introduced  by  the 
presence  of  the  higher.  And  so  the  whole  would  be  more  easily  dis- 
tinguishable from  the  pure  tone  or  from  one  tone  than  would  the  octave, 
whether  the  two  pitches  were  recognised  in  the  whole  or  not.    By  the 

2—2 


20  DEGREES  AND  THEORY  [ch. 

unpractised  person  the  two  pitches  would  be  more  easily  recognisable 
than  they  are  in  the  octave;  for  the  musical  mind,  there  would  be  less 
interpenetration  of  the  two  tones  although  he  might  not  be  able  to 
detect  any  difference  in  the  ease  with  which  the  two  tones  were  recognised ; 
to  both  there  would  be  less  balance  or  smoothness  in  the  whole  sound 
than  in  the  case  of  the  octave.  We  may  therefore  identify  this  case 
with  the  second  grade  of  fusion — the  fifth. 

And  we  are  confirmed  in  so  doing  by  the  well  known  difference 
between  the  octave  and  fifth — namely  that  octaves  are  simultaneously 
compatible  with  one  another,  whereas  fifths  are  not.  If  we  add  a  double 
octave  to  the  first,  the  third  pitch-point  falls  exactly  into  the  upper 
half  of  the  middle  tone  without  disturbing  the  relative  balance  of  the 
two  lower  tones ;  but  if  we  add  a  second  fifth,  it  will  not  thus  fit  in  with 
the  first  two  tones.  On  the  contrary,  it  entirely  spoils  the  balance  and 
symmetry  of  the  first  fifth. 

It  follows  from  the  nature  of  the  balance  claimed  for  the  fifth  that 
the  volumes  of  its  two  tones  are  as  3  to  2.  And  along  with  the  octave 
case  this  implies  that  the  volumes  of  all  tones  that  have  the  ordinary 
fusional  relations  to  one  another,  are  inversely  proportional  to  the 
corresponding  rates  of  physical  vibration.  But  the  relations  of  the 
volumes  have  been  educed  without  any  appeal  to  this  famihar  physical 
fact,  at  least,  logically.  So  long  as  this  is  so,  it  is  a  matter  of  impossible 
speculation  to  discover  whether,  given  the  proper  analytic  approach 
to  the  problem,  the  volumic  ratios  would  have  been  discovered  and 
sufficiently  proved,  had  the  physical  knowledge  not  preceded.  Our  only 
concern  need  be  whether  sufficient  ground  now  exists  upon  which  to 
raise  a  logically  independent  proof.  This  undertaking  is  not  a  piece  of 
mere  pedantry,  as  some  might  think  who  recall  that  one  way  of  proving 
a  thing  is  enough.  For  the  proof  given  has  not  for  its  object  the  re- 
proving of  truths  concerning  physical  ratios,  but  the  demonstration  of 
an  entirely  new  object,  namely  the  ratios  of  the  volumes  of  tones, 
solely  as  they  are  heard.  This  object  is  a  psychological  one,  if  you  like. 
It  deals  with  an  object  that  is  as  different  from  physical  vibrations  as 
blue  is  from  commotions  of  the  ether  or  as  thought  is  from  brain  process. 

It  is  as  absurd  to  suppose  that  the  only  objects  we  can  study 
logically  and  scientifically  and  convincingly  are  physical  as  it  would  be 
to  suggest  that  we  could  not  think  correctly  until  we  knew  the  physiology 
of  the  brain  perfectly.  You  may  reply  that  of  course  we  think  correctly 
because  the  brain  is  there  working  correctly  whether  we  know  of  its 
workings  or  not.    True;  and  when  we  prove  things  properly  about 


in]  OF  CONSONANCE  AND  DISSONANCE  21 

tones,  the  brain  doubtless  works  properly  too.  But  the  chief  interest 
for  us  is  to  think  correctly  first.  There  is,  of  course,  no  doubt  that  the 
brain  must  be  capable  of  interacting  in  some  special  way  with  physical 
sounds  if  we  are  to  hear  sounds  and  to  think  about  them  correctly. 
But  experience  is  more  than  brain.  And  men  may  yet  have  to  infer 
something  from  the  study  of  sounds  as  they  are  for  us  when  heard,  if 
they  are  to  understand  how  the  brain  acts  in  connexion  with  them. 
Besides,  brain  and  mind  or  experience,  or  more  specifically,  brain  and 
tones  or  music  are  two  very  distinct  and  different  things  that  no  one 
could  possibly  confuse  with  one  another,  no  matter  how  much  they 
may  be  dependent  upon,  and  may  interact  with  one  another.  So  each 
of  them  must  be  studied  for  its  own  sake  and  in  the  special  way  its 
peculiar  nature  and  its  relation  to  ourselves  make  possible.  We  cannot 
bring  our  knowledge  of  both  into  harmony  until  we  have  attained  a 
complete  knowledge  of  each. 

The  form  of  overlapping  of  the  other  intervals  and  the  kind  of 
balance  between  the  parts  of  the  volume  they  constitute  may  easily 
be  reckoned  out  from  the  knowledge  we  have  just  gained.  In  the  case 
of  the  fourth,  the  higher  volume  will  be  three-quarters  of  the  length 
of  the  lower  volume.  So  the  lower  limit  of  the  upper  tone  will  fall  exactly 
half-way  between  the  lower  limit  of  the  lower  tone  and  its  pitch-point. 
And  the  pitch-point  of  the  upper  tone  being  at  its  centre  will  lie  three- 
eighths  of  the  length  of  the  lower  tone  from  its  upper  terminal  and  one- 
eighth  from  the  pitch-point  of  the  lower  tone.  The  whole  mass  of  sound 
of  the  two  tones  will  be  marked  into  parts  of  two,  two,  one,  and  three, 
eighths  of  its  length.  There  is  here  some  balance  in  the  one  half  of  the 
whole  volume  and  less  in  the  other.  For  the  major  third  the  parts 
are  2,  3,  1,  and  4  tenths,  which  seem  more  irregular. 

As  we  proceed,  the  disproportion  of  the  parts  seems  to  grow  greater 
and  one  or  other  of  the  points  of  the  higher  tone  falls  nearer  and  nearer 
to  the  pitch  of  the  lower  one.  In  so  doing  it  must,  of  course,  become 
more  noticeable;  just  as  we  catch  sight  of  a  second  visual  point  the 
more  easily  the  nearer  (within  ordinary  Umits)  it  lies  to  another  that 
we  notice  easily  and  are  attending  to.  We  notice  the  pitch-point  of 
the  lower  tone  more  easily  because  it  lies  in  the  centre  of  the  whole 
volume  of  sound  formed  by  the  two  tones;  and  as  pitch  is  always 
central  in  tone  and  we  are  constantly  at  work  with  the  pitches  of  tones 
in  music,  so  we  get  into  the  habit  of  attending  to  tones  in  a  central 
balanced  manner  and  then  notice  the  pitch  and  volume  of  the  lowest 
component  of  a  chord  most  easily.   We  do  so,  not  only  when  the  chord 


22  THEORY  OF  CONSONANCE  [ch.  m 

is  an  isolated  stationary  mass  of  sound,  but  also  in  music  where  each 
tone  is  a  phase  of  a  voice  or  part,  except  in  so  far  as  some  melodic 
figure  or  theme  is  present  that  specially  makes  out  another  element 
than  the  lowest  one  of  the  moment. 

But  a  study  of  the  proportions  of  the  parts  of  the  lower  fusions 
shows  no  great  differences  between  them.  That  accords  well  with  the 
slight  differences  they  show  for  the  ear.  And  there  seems  to  be  no 
ready  way  in  which  we  might  make  our  conceptual  or  theoretical 
treatment  of  these  volumic  parts  give  us  a  much  more  exact  account 
of  the  degrees  of  fusion  of  intervals  than  the  ear  gives,  so  as  to  provide 
a  rule  or  rapid  guide  to  the  ear,  as  e.g.  measurement  is  for  many  purposes 
a  guide  and  control  to  the  eye.  That  deficiency  does  not  indicate  that 
the  volumic  theory  of  fusion  is  wrong.  The  agreement  the  theory 
shows  between  concept  and  sound,  speaks  only  in  its  favour.  Measure- 
ment is  by  no  means  an  infallible  guide  or  a  standard  of  visual  art. 
It  is  possible,  however,  that  calculation  may  yet  discover  more  delicate 
or  more  adequate  ways  of  representing  the  volumic  basis  of  fusion, 
and  from  following  and  moulding  itself  to  portray  the  verdicts  of  hearing, 
may  gain  strength  to  place  itself  ahead  of  the  sense  and  to  lead  it  for- 
wards to  unexplored  regions  where  it  may  dwell  with  pleasure  and 
where  its  art  may  flourish  more  abundantly.  If  that  is  possible,  it  would 
be  foolish  not  to  cherish  the  idea  merely  because  theory  has  rarely, 
if  ever,  yet  preceded  the  experiments  of  musical  art.  On  the  contrary 
we  have  every  reason  to  hope  that  theory  will  yet  be  as  great  a  support 
to  art  as  it  has  come  to  be  to  industry.  And  we  shall  do  well  to  build 
hopes  for  art  upon  this  dream.  After  all,  such  theory  does  not  wish  to 
import  into  art  influences  and  notions  entirely  extraneous  to  its  matter, 
as  are  ratios  of  vibrations  and  all  merely  physical  knowledge.  At  its 
best  such  theory  is  really  only  a  most  perfect  description  of  the  actual 
things  that  the  art  of  music  deals  with,  tones  and  their  groupings.  It 
gives  the  artist  a  better  comprehension  of  their  real  being  than  he 
would  gain  from  his  ear  alone.  It  is  merely  the  hearing  of  the  ear 
perfected  and  purified  by  the  wide  attention  and  analysis  of  the  intellect. 
If  the  art  of  music  can  turn  such  work  of  the  intellect  to  fruitful  use, 
why  should  it  not  be  allowed  to  profit  thereby?  After  all  men  have 
had  ears  and  eyes  since  the  dawn  of  time;  but  they  have  not  always 
had  the  minds  to  make  art  out  of  sounds  and  sights.  Their  minds 
have  had  to  grow  to  the  power  of  that  creation. 


CHAPTER  IV 

THE  RELATIONS  OF  FUSION  TO  BEATS,  PARTIALS,  AND 
DIFFERENCE  TONES 

Having  attained  what  seems  to  be  a  satisfactory  explanation  of  fusion, 
one  that  is  grounded  upon  the  fusing  tones  themselves  and  does  not 
involve  any  reference  to  any  other'  phenomena  that  may  accompany 
the  simultaneous  occurrence  of  two  tones,  we  may  feel  free  from  any 
obhgation  to  refute  those  theories  that  base  their  explanation  of 
consonance  solely  upon  such  adventitious  phenomena.  Besides  these 
theories  have  already  been  well  refuted  (67);  so  that  before  a  theory 
of  consonance  by  volumic  balance  had  become  attainable,  the  dis- 
cussion of  the  problem  had  yielded  the  conclusion  that  the  grades  of 
fusion  are  an  immediate  characteristic  of  pairs  of  simultaneous  tones 
and  that  the  most  probable  explanation  of  these  grades  was  to  be 
sought  in  some  form  of  sensory  'synergy'  (64,  214).  An  analogous  case 
is  famihar  in  vision,  where  certain  colours  are  found  to  look  well  together, 
while  others  'kill'  their  neighbours.  We  have  no  very  satisfactory 
explanation  of  these  pecuhar  harmonies  of  vision ;  but  the  most  probable 
theory  supposes  that  the  physiological  processes  underlying  in- 
harmonious or  discordant  colours  in  the  retina  or  in  the  central  areas 
of  the  brain  that  subserve  vision,  do  not  work  easily  together  or  do 
not  make  material  for  each  other,  as  it  were,  or  predispose  each  other's 
functions.  The  notion  of  'synergy'  is,  then,  not  a  specific  theory  so 
much  as  an  attempt  to  indicate  where  the  basis  of  a  true  theory  will 
probably  be  found.  And  the  volumic  theory  given  above  may  quite 
well  be  looked  upon  as  a  solution  of  this  query  as  to  the  exact  nature 
of  fusional  'synergy.' 

This  by  no  means  recent  (1893)  reduction  of  the  field  in  which  the 
explanation  of  fusion  may  be  sought  seems  to  be  still  unknown  to  many 
of  those  who  are  interested  in  the  theoretical  foundations  of  music^. 

*  Thus,  for  example,  in  the  year  1917  Shirlaw  (60,  48i)  writes:  "The  only  thing  which 
theorists  who  have  made  the  harmonic  series  the  principle  of  chord  generation  appear 
to  have  omitted  to  do  has  been  to  abide  by  the  results  of  their  own  theory.  Having 
accepted  a  fundamental  and  guiding  principle  of  harmony,  they  have  nevertheless  refused 
to  be  guided  by  it,  and  have  virtually  abandoned  it,  or,  while  still  professing  to  do  it 
homage  have  vainly  attempted  to  exploit  it  for  their  own  purposes.  The  principle  of 
harmony  of  Zarlino,  Descartes,  Rameau,  Tartini,  furnishes  us  with  but  a  single  chord. 


24  THE  RELATIONS  OF  FUSION  [ch. 

It  may  therefore  be  profitable  to  make  a  short  review  of  the  critical 
part  of  this  work  now.  In  so  doing  we  shall  best  follow  the  exposition 
given  by  the  author  of  the  theory  of  'synergy.'  We  shall  thereby  not 
only  do  justice  to  an  important  phase  of  the  development  of  the  science 
of  sound,  but  we  may  also  reach  a  greater  completeness  of  scientific 
outlook.  For  it  is  quite  possible  that  once  the  volumic  basis  of  fusion 
is  given,  its  nature  may  be  heightened  or  lessened  by  these  other 
adjuncts  of  the  fusing  intervals,  even  although  they  are  incapable  of 
producing  by  themselves  the  effect  of  consonance  or  dissonance. 

The  most  familiar  and  the  most  widely  accepted  theory  of  consonance 
is  that  of  Helmholtz.  Helmholtz  gives,  as  Stumpf  (67,  2)  pointed  out, 
"not  one,  but  two  different  definitions  of  consonance  and  dissonance, 
which  are  indeed  closely  interwoven  in  his  exposition,  but  which  really 
are  quite  different  and  apply  to  different  fields."  According  to  Heffernan^ 
(19)  there  are  indeed  no  less  than  three  elements  in  Helmholtz's  explana- 
tions of  consonance. 

The  chief  definition  is:  "consonance  is  a  continuous,  dissonance  an 
intermittent  sensation  of  tone"  (20,226).  The  intermittence  is  here 
created  by  the  beats  which  appear  when  tones  of  about  the  same  number 
of  vibrations  per  second  occur  together.  As  far  as  they  are  audible, 
the  number  of  beats  is  equal  to  the  difference  between  the  numbers 
of  the  two  rates  of  vibration.  These  beats  may  originate  from  the 
primary  tones  (as  in  the  interval  of  a  minor  second)  or  from  the  upper 
partials  of  the  primaries  (as  when  the  fifth  partial  e^  of  c^  beats  with 
the  fourth  partial  /^  of  f^)  or  from  a  partial  of  the  one  tone  with  the 
other  primary  (as  when  c^  beats  with  the  second  partial  of  b^);  it  is 
a  matter  of  indifference  what  their  source  is,  so  long  as  they  make 
the  binary  sound  noticeably  rough.  For  they  must  then  certainly  help 
to  make  dissonances  less  suitable  for  pleasure  and  for  the  clear  exposition 

But  this  ought  not  to  be  regarded  as  a  negative  result,  but  as  a  positive  result  of  the 
greatest  theoretical  significance.  It  is  the  one  fact  of  supreme  importance  which  this 
principle  has  to  teach  us.  This  has  not  yet  been  realised....  There  exists  in  our  harmonic 
music  but  a  single  chord,  from  which  all  others  are  developed.  But  as  the  sounds  of  this 
harmony  are  contained  in  the  resonance  of  musical  sound  itself,  all  harmony  has  its  source 
in  a  single  musical  sound.  The  development  of  harmony  has  been  a  more  simple  and 
beautiful  process  than  musicians  and  theorists  have  imagined."  And  in  a  footnote  to  the 
word  "developed"  Shirlaw  promises  "a  new  and  smaller  constructive  work  on  the  theory 
of  harmony." 

^  This  paper  (1887)  gives  a  striking  criticism  of  Helmholtz,  but  its  experimental 
basis  is  at  times  insufficient  and  lacking  in  precision. 


IV]      TO  BEATS,  PARTIALS,  AND  DIFFERENCE  TONES       25 

that  art  requires  than  they  would  otherwise  be.  And  if  beatless  sounds 
are  of  themselves  smooth,  contrast  with  the  roughness  of  beating 
chords  will  make  them  seem  smoother  and  more  beautiful  still. 

But  it  is  important  to  notice  that  Helmholtz's  theory  can  by  no 
means  justify  the  ascription  of  smoothness  to  beatless  chords.  Smooth- 
ness is  not  a  mere  negation;  a  table  is  not  smooth  to  touch  as  soon 
as  it  ceases  to  be  rough;  it  is  smooth  only  so  long  as  it  gives  the  finger 
a  continuous  area  of  unvaried  sensation.  And  the  regularity  and  positive 
smoothness  we  have  shown  to  exist  in  tone  make  any  such  purely 
negative  use  of  smoothness  inadmissible.  Still  it  is  evident  that  the 
roughness  of  beats  will  heighten  and  increase  the  general  effect  of  the 
asymmetry  and  unbalanced  disproportion  of  parts  that  we  have  shown 
to  constitute  the  primary  being  of  dissonance.  It  will  make  differences 
and  grades  of  unsuitability  and  unpleasantness  amongst  the  lower 
grades  of  fusion  that  might  otherwise  be  less  distinctive. 

So  far  the  theory  would  thus  account  only  for  grades  of 
unpleasantness  amongst  combinations  of  simultaneous  tones.  For 
successive  tones  the  explanation  fails  altogether,  since  no  beats  then 
appear  in  any  case.  Here  Helmholtz  appeals  to  his  second  basis  of 
consonance  in  the  partials  that  are  common  to  the  successive  tones. 
In  the  octave  we  hear  again  the  largest  possible  part  of  what  we  heard 
before  in  the  lower  tone,  namely  every  even  numbered  partial.  In 
the  fifth  we  hear  a  lesser  part,  but  still  much,  namely  every  partial 
that  is  a  multiple  of  three.  And  so  on  (20,  253  ff.).  We  do  not  need  to 
have  analysed  these  partials  from  the  whole  tone  by  special  attention 
and  to  know  of  them.  The  second  tone  merely  appears  to  be  like  the 
first  one  according  to  the  amount  of  identity  amongst  the  partials  of 
the  two,  as  two  faces  often  look  similar  without  our  being  able  to  say 
in  what  respect  precisely  they  are  so. 

This  second  theory  is  in  a  notable  respect  the  obverse  of  the  first, 
as  it  were.  For  now  the  positive  quality  belongs  to  consonance  alone. 
Dissonance  is  a  mere  negation  of  consonance;  two  tones  are  dissonant 
when  they  have  nothing  in  common ;  or  dissonance  is  the  lowest  degree 
of  consonance  and  yet  it  is  not  consonance  at  all.  Moreover  it  is  clear 
that  the  continuity  thus  established  between  successive  tones  would 
be  a  real  influence  connecting  them,  in  so  far  as  it  could  be  felt  in  the 
aggregate;  and  though  it  would  necessarily  of  itself  be  only  a  weak 
bond  between  tones,  it  would  undoubtedly  come  to  the  good  of  any 
obvious  bond  already  linking  consonant  tones  to  one  another. 

But  Helmholtz's  second  theory  does  not  apply  to  simultaneous 


26  THE  RELATIONS  OF  FUSION  [ch. 

tones.  For  the  partials  that  are  common  to  two  tones  an  octave  apart 
could  at  the  most  only  constitute  a  set  of  further  primary  tones  to  the 
two  from  which  they  originate.  Of  course  they  do  not  under  ordinary 
circumstances  do  anything  of  the  sort.  Besides,  Helmholtz's  theory 
is  generally  quite  unable  to  explain  satisfactorily  why  a  tone  and  its 
partials  are  heard  as  a  unit  and  not  as  a  number  of  tone  spots  or  separate 
tones. 

"We  have  thus  in  fact  two  different  principles  in  Helmholtz's  theory, 
the  one  valid  exclusively  for  simultaneous,  the  other  exclusively  for 
successive  tones.  This  state  of  things  seems  strangely  to  have  escaped 
his  notice,  and  as  he  himself  nowhere  emphasised  the  twofold  nature 
of  his  definition,  it  has  also  generally  and  from  the  first  not  been  felt 
to  be  a  defect"  (67,  4;  cf.  34,  160).  Such  a  duplication  and  crossing 
of  explanatory  principles  is  certainly  derogatory  to  any  theory  of  the 
system  of  consonances  and  kindred  relations  in  successive  and  simul- 
taneous tonal  groups.  That  system  nowhere  suggests  any  such  dual 
nature;  and  we  could  hardly  expect  two  such  different  causes  to  yield 
so  homogeneous  a  system  of  phenomena.  In  face  of  the  positive  theory 
of  consonance  given  above  these  logical  and  phenomenal  inconsistencies 
would  be  enough  to  refute  Helmholtz's  theories  alone.  But  the  following 
arguments  have  been  stated  besides  (67,  4ii.). 

As  beating  is  a  periodical  change  of  tonal  intensity,  such  inter- 
mittence  of  sensation  can  easily  be  produced  either  in  a  single  tone, 
or  in  each  of  two  simultaneous  tones.  A  sounding  instrument  may, 
for  example,  be  placed  in  a  closed  box  from  which  a  tube  leads  to  the 
surface  of  a  rotating  disc;  in  this  disc  as  many  holes  as  need  be  are 
pierced,  so  that  they  pass  the  mouth  of  the  tube  when  the  disc  is  rotated. 
Dissonance  does  not  then  appear  any  more  than  it  does  when  a  consonant 
interval  is  played  as  a  tremolo.  Besides  such  beats  without  dissonance, 
we  can  have  dissonance  without  beats,  as  from  tuning-forks  on  their 
resonance  boxes  sounding  to  500  or  490  and  to  700  vibrations,  or  to 
700  and  1000,  or  to  780  and  1100.  Such  forks  well  sounded  contain 
hardly  more  than  a  trace  of  the  first  partial  and  at  these  rates  of  vibration 
beats  are  quite  inaudible.  Moreover  when  forks  without  their  resonators, 
preferably  between  800  and  1200  vibrations,  are  held  one  before  each 
ear,  their  tones  are  not  carried  to  the  opposite  ears,  either  by  the  air 
or  by  the  bones  of  the  head,  so  long  as  they  are  not  made  too  loud. 
When  two  forks  of  say  800  and  900  (a  major  tone)  are  tested  in  this 
way,  only  a  trace  of  beating  can  be  heard  at  the  very  most;  in  striking 
contrast  to  the  effect  of  both  forks  before  a  single  ear;  but  the  dissonance 


IV]      TO  BEATS,  PARTIALS,  AND  DIFFERENCE  TONES       27 

in  the  two  cases  remains  the  same.  When  in  turn  a  consonance,  e.g.  a 
major  third  (620  and  775  vibrations),  is  tested  thus,  it  remains  just  as 
fused  as  usual. 

The  number  of  beats  accompanying  any  dissonant  interval  (e.g.  a 
major  second)  must  gradually  increase  as  the  interval  is  raised  through 
the  musical  range  of  pitch;  but  there  is  no  indication  that  it  therefore 
becomes  a  dissonance  only  at  a  certain  pitch  or  ceases  to  be  a  dissonance 
in  the  higher  octaves.  Nevertheless  the  roughness  that  is  due  to  beating 
and  that  accompanies  dissonance  under  ordinary  circumstances  will 
indeed  vary  with  the  audibility  of  the  beating  (which  varies  with  its 
rate).  Nor  do  differences  of  timbre  make  such  regular  differences  in 
the  degree  of  dissonance  or  in  the  musical  usage  of  intervals  as  they 
might  be  expected  to  do  in  view  of  the  different  possibilities  of  beating 
they  create. 

But  it  is  needless  to  follow  the  argument  further.  Let  us  consider 
the  alternative. 

The  chief  argument  against  consonance  by  coincidence  of  partials 
is  its  continued  appearance  amongst  tones  devoid  of  partials.  And, 
Helmholtz's  work  on  timbre  shows  that  the  different  musical  instru- 
ments, far  from  having  each  the  complete  series  of  possible  partials, 
differ  precisely  in  the  selection  from  the  series  that  is  typical  of  them; 
and  yet  the  grades  of  consonance  do  not  differ  from  instrument  to 
instrument  or  from  one  intonation  to  another.  Suppose,  for  instance, 
that  the  clarinet  has,  as  Helmholtz  says^,  only  the  uneven  partials; 
then  an  octave  on  clarinets  could  not  possibly  be  a  consonance  at  all, 
but  rather  an  extreme  dissonance,  because  in  the  second  tone  we  should 
hear  nothing  of  what  we  heard  in  the  first.  By  means  of  'interference' 
we  can  exclude  from  a  tone  any  specified  partials  so  as  to  make  coinci- 
dence unattainable.  Consonance,  however,  remains  unaffected.  The 
consonance  of  tuning-fork  tones  is  beautiful  in  spite  of  the  restriction 
of  their  partials.  Helmholtz  may  well  have  been  aware  of  the  main- 
tenance of  consonance  in  spite  of  the  relative  purity  of  fork  tones. 
However  he  may  have  adjusted  his  mind  to  this  fact,  his  successors 
at  least  have  variously  appealed  to  the  influence  of  memory.  But, 
as  Stumpf  says  (67,16):  "The  remembrance  that  two  other  blends 
on  the  same  fundamentals  once  were  consonant,  can  only  bring  the 

^  According  to  D.  C.  Miller's  harmonic  analyses  the  seventh,  eighth,  ninth  and  tenth 
partials  predominate  in  the  blend  of  the  clarinet.  "The  seventh  partial  contains  eight 
per  cent,  of  the  total  loudness,  while  the  eighth,  ninth  and  tenth  contain  18,  15  and  18 
per  cent,  respectively"  (40,  aoi).   The  second,  fourth  and  sixth  partials  are  veiy  weak. 


28  THE  RELATIONS  OF  FUSION  [ch. 

non-consonance  of  the  present  tones  by  contrast  more  strongly  to  my 
notice.  A  dish  that  lacks  salt  would  never  be  said  to  be  well  salted  by 
mere  force  of  memory  or  custom;  on  the  contrary." 

"  In  short,  timbre  is  for  one  and  the  same  interval  extremely  variable, 
but  the  degree  of  consonance  is  constant.  Hence  both  cannot  be 
explained  from  one  and  the  same  principle.  And  it  is  just  the  happy 
explanation  of  timbre  that  Helmholtz  achieved  for  acoustics  for  all 
time  that  makes  his  explanation  of  consonance  from  the  same  principle 
an  impossibility"  (67,  19). 

Various  attempts  have  been  made  to  bring  consonance  and  its 
grades  into  relation  to  another  class  of  phenomena  that  accompany 
the  simultaneous  occurrence  of  two  or  more  tones,  namely  difference 
tones.  The  earliest  such  attempt  was  made  by  their  discoverer  ^  Tartini 
after  whom  they  were  often  called  '  Tartinian  tones.'  Helmholtz  appealed 
to  them  in  explanation  of  the  less  harmonious  effect  of  the  minor,  as 
compared  with  the  major,  triad,  thus  introducing  a  third  factor  in  the 
creation  of  consonance.  Of  recent  years  an  elaborate  attempt  has  been 
made  to  base  the  grades  of  consonance  solely  upon  the  beating  and 
confusion  (of  a  special  kind)  of  neighbouring  difference- tones.  On  the 
basis  of  numerous  observations  of  the  difference-tones  that  accompany 
intervals  of  different  ratios,  it  was  claimed  that  the  greater  dissonance 
had  the  greater  number  of  difference-tones  within  close  pitch-distance 
of  one  another,  and  would  therefore  have  the  more  beating  and  confused 
blurring.    That  is  to  say  the  latter  constitute  dissonance^. 

This  type  of  theory  seems  to  escape  the  criticism  fatal  to  Helmholtz's 
explanation  by  coincidence  of  partials, — that  its  basis  is  withdrawn 
when  the  primary  tones  are  purified  of  all  partials.  For  difference-tones 
still  accompany  such  pure  tones.  They  are  not  due,  like  partial  tones, 
to  any  physical  process  in  the  sonorous  body  or  in  the  air  between  that 
and  the  ear,  but  they  arise  somewhere  within  the  ear  directly  from  the 

^  Or  'one  of  their  discoverers.'  Cf.  60,  30i:  "Although  Tartini  is  generally  regarded 
as  the  first  to  discover  the  combination  tones — he  had  asserted  that  as  early  as  1717  he 
had  made  use  of  them  for  the  purpose  of  teaching  pure  intonation  on  the  violin  to  his 
pupils — it  is  certain  that  other  musicians  had  discovered  them  independently.  J.  A.  Serre 
of  Geneva,  and  Romieu  of  Montpellier,  had  given  accounts  of  these  tones  before  Tartini's 
publication  of  the  Trattato  di  musica  (1754)."  G.  A.  Sorge  in  his  Vorgemach  der  musi- 
kalischen  Komposition,  "pubhshed  nine  years  before  Tartini's  Trattato  di  musica,  demon- 
strates his  acquaintance  with  the  phenomenon  of  combination  tones." 

*  For  sources  and  criticism  see  68,  57.  For  the  most  trustworthy  and  complete  record 
of  observations  of  difEerence-tones  see  70. 


IV]      TO  BEATS,  PARTIALS,  AND  DIFFERENCE  TONES       29 

interaction  of  the  primaries.  But  difference-tones  can  be  greatly 
weakened,  if  not  made  to  disappear  entirely  when  the  primary  tones 
are  presented  one  to  each  ear  and  are  given  in  somewhat  weak  strength 
(cf.  7).  The  dissonance,  however,  does  not  then  disappear  nor  change 
its  degree.  Besides  it  is  a  fatal  defect  of  this  type  of  theory  that  it  gives 
no  really  positive  status  and  explanation-  to  consonance.  Consonance 
is  here  a  mere  negation  or  minimum  of  dissonance.  And  whatever  be 
the  nature  and  cause  of  dissonance  that  is  postulated — whether  it 
consist  in  the  multiplicity  of  the  difEerence-tones  or  in  the  fluctuations 
of  their  beating  or  whether  it  be  traceable  rather  to  the  confused 
indistinguishability  of  too  closely  neighbouring  difference-tones  that 
form  between-tones  or  even  a  sort  of  streak-tone,  or  the  like — we 
should  in  any  case  certainly  have  no  reason  to  hear  non-dissonant 
sets  of  tones  in  any  other  way  than  with  the  greatest  precision  and 
clearness  of  distinction  from  one  another  (cf.  68,  282).  Even  though, 
as  in  the  great  consonances,  the  number  of  difference-tones  is  greatly 
reduced  (to  none  in  the  octave),  the  two  primary  tones  would  still  be 
two  in  all  obviousness;  there  could  be  no  excuse  for  holding  them  to 
be  but  one,  and  no  ground  for  establishing  any  special  relation  between 
them  (such  as  that  of  'consonance')  except  that  of  clear  distinguish- 
ability.  Thus  the  appeal  to  difference- tones  can  only  give  a  partial 
explanation  and  must  therefore  be  unsatisfactory. 

If,  however,  a  positive  explanation  of  consonance  and  dissonance 
in  their  grades  has  already  been  given,  as  in  the  previous  pages,  it  is 
obvious  that  any  beating  of  difference-tones  amongst  one  another  or 
with  the  other  components  of  the  whole  sound  would  add  to  the  rough- 
ness and  irregularity  inherent  in  the  latter  through  its  primary  com- 
ponents, while  the  consonance  of  these  parts — which  would  have  to 
rest  upon  the  same  kind  of  processes  as  the  consonance  of  the  primaries, 
— would  bear  out  the  latter.  Consider,  for  example,  the  case  of  the 
perfect  fifth  in  pure  tones  in  relation  to  the  two  loudest  difference- 
tones — the  'first'  (h-l)  and  the  'second'  {2l-h).  The  ratio  for  the 
fifth  is  2  :  3.  Its  difference- tones  are  both  of  ratio  1.  A  slight  mistuning 
will  yield  one  difference-tone  just  less  than  1  and  another  just  more 
than  1.  These  two  wiU  beat  with  one  another,  whereas  in  the  just 
interval  we  shall  have  three  consonant  intervals,  octave,  fifth  and 
twelfth.  Similarly  in  the  two  common  triads,  major  and  minor,  whose 
ratios  are  4:5:6  and  10  :  12  :  15  respectively,  we  find  the  following 
components  :  in  the  major  chord,  1  (twice),  2  (twice),  3,  4,  4,  5,  6,  or 
C^,  c,  g,  c^,  e^,g^;  in  the  minor  chord,  2,  3,  5  (twice),  8,  9,  10,  12,  15,  or 


30  PARTIALS  AND  DIFFERENCE  TONES  [ch.  iv 

A\^,  E\f,  c,  a|7,  b\^,  c^,  e^\fy  g^.  Apart  from  the  difference  of  octaves 
there  are  three  dissonant  intervals  in  the  latter  chord  and  fewer  high 
grade  consonances  (3  octaves,  6  fifths,  and  2  fourths  to  4  octaves, 
5  fifths,  and  1  fourth).  We  shall  return  to  this  topic  again  (p.  192  £.), 
The  general  series  of  the  upper  partials  and  the  difference-tones 
were  on  the  whole  a  relatively  late  discovery  in  the  history  of  the 
scientific  foundations  of  music.  But  many  years  before  Helmholtz 
propounded  his  very  convincing  theory  of  instrumental  tone-blend 
(timbre),  they  had  become  familiar  to  all  the  leading  theorists.  If  any 
feature  at  all  of  tones  were  really  explicable  in  terms  of  some  such 
adventitious  accompaniments  of  primary  tones,  or  rather  consisted  of 
them,  we  might  certainly  have  expected  Helmholtz's  predecessors  to 
have  learned  how  to  explain  pitch-blend  by  the  grouping  of  partials. 
That  they  did  not  do  so,  and  obviously  were  not  tempted  to  do  so, 
gives  us  the  right  to  consider  it  highly  improbable,  apart  from  all  other 
grounds,  that  so  direct  and  unmistakable  phenomena  as  those  of 
consonance  and  dissonance  are  founded  upon  such  remote  accompani- 
ments of  primaries  as  partials  and  difference-tones  and  their  beatings. 
We  must  find  the  basis  of  consonance  and  dissonance,  as  it  were, 
directly  in  or  below  the  primaries  themselves.  And  that  the  theory 
propounded  above  has  succeeded  in  doing. 


CHAPTER  V 

THE  CONSONANCE  OF  SUCCESSIVE  TONES 

We  have  not  yet  given  an  account  of  the  consonance  of  successive 
tones  from  the  standpoint  of  the  volumic  theory.  But  it  is  evident 
that  the  task  is  a  very  simple  one  and  involves  no  change  of  the  basis 
of  explanation  and  no  new  principle.  This  necessity  characterises  all 
the  other  theories  we  have  noticed.  There  is  no  beating  between 
successive  tones,  so  that  there  can  be  no  roughness  between  them. 
And  while  one  tone  may  certainly  repeat  a  number  of  the  partials  of 
a  preceding  one,  yet  there  is  no  means  of  detecting  which  of  the  partials 
appearing  with  two  simultaneous  primaries  belong  to  either,  in  so  far 
at  least  as  they  might  belong  to  both.  Finally,  the  difference-tones 
that  appear  with  simultaneous  tones  are  lacking  in  their  sequence. 

The  attempt  has  been  made  to  cover  over  these  lapses  of  the  basis 
of  explanation  by  appeal  to  the  restorative  work  of  memory!  The  idea 
is  that  when  the  basis  of  consonance  or  dissonance  is  actually  given, 
the  memory  will  mark  it  well  and  associate  it  with  the  primary  tones 
which  it  accompanies.  Later  when  these  primaries  appear  without 
.the  basis  of  their  fusion,  this  characteristic  will  be  restored  by  the 
memory  and  the  primaries  will  function  as  fused.  There  is  no  general 
psychological  fault  in  this  theory  so  far  as  the  memory's  activity  is 
concerned.  Seeing  a  man  often  and  hearing  him  speak,  we  learn  to 
connect  his  voice  with  his  visual  appearance;  when  we  later  merely 
see  bim  we  can  call  his  voice  vividly  to  mind.' 

But  we  do  not  then  hear  him  speak.  Memory  does  not  cause  hallu- 
cinations in  the  most  of  us,  nor  should  we  desire  it  to  do  so.  There 
are,  however,  cases  of  a  less  abnormal  character  which  seem  to  imply 
a  restoration  of  sensation  by  memory.  Thus  a  glowing  iron  is  often 
said  to  look  hot,  a  child's  cheek  looks  soft  and  tender,  the  ground  after 
rain  looks  wet.  True;  but  these  things  do  not  then  feel  hot  or  tender 
or  wet;  they  merely  look  so,  because  their  visual  appearance  makes  us 
think  at  once  of  the  associated  character  that  comes  through  the  other 
sense.  There  is  a  certain  visual  feature  in  each  which  prompts  the 
mind  to  recall  the  associate,  and  so  that  visual  feature  acquires  in  our 
minds  a  special  meaning  as  a  sign  of  the  associate.  But  the  visual 
'ground'  no  more  feels  wet  because  it  looks  so,  than  the  word  'lead' 


32  THE  CONSONANCE  OF  SUCCESSIVE  TONES  [ch. 

acquires  a  weight  because  it  is  the  sign  of  a  heavy  thing.  On  the  con- 
trary when  a  thing  looks  heavy,  it  usually  feels  lighter  than  another 
thing  of  the  same  weight  that  does  not  look  heavy  or  that  looks  light. 
This  is  the  size- weight  illusion,  demonstrable  with  two  objects  of 
equal  weight  but  of  different  size.  Expecting  weight  we  do  not  feel  it 
to  be  greater  than  usual,  but  less. 

At  least  as  much  as  this  is  also  true  of  tones  in  relation  to  consonance. 
If  you  hear  c  and  then  d  and  recognise  the  interval  between  them 
specifically  as  a  whole  tone  or  merely  as  much  smaller  than  the  consonant 
intervals,  you  may  certainly  recall  the  fact  that  these  tones  together 
would  form  a  dissonance.  But  they  would  not  therefore  sound  dissonant 
in  sequence.  On  the  contrary,  if  you  had  learnt  from  the  simultaneous 
tones  to  expect  a  dissonance,  then  on  hearing  them  in  sequence  you 
would  be  greatly  struck  by  the  absence  of  dissonance:  just  as  you  would 
be  astonished  by  the  weight  of  a  cigarette  you  had  picked  from  a  box 
if  it  happened  to  be  made  of  lead. 

The  suggested  explanation  of  fusional  degrees  by  'synergy'  has 
met  with  a  similar  difficulty  in  explaining  the  relations  of  successive 
tones  to  consonance.  A  solution  by  presumption  offers  itself  readily 
enough,  however.  For  we  may  suppose  that  the  special  function  by 
which  two  tones  make  each  other's  action  easier  or  harder  when 
simultaneous,  still  exists  when  they  are  successive.  For  the  earlier 
tone  is  not  then  entirely  gone,  any  more  than  the  earlier  part  of  a  melodic 
phrase  is  mentally  non-existent  when  the  later  notes  are  being  played. 
The  function  of  fusion  would  then  hold  between  tones  that  are  *  together 
in  the  mind,'  so  to  speak,  whether  simultaneously  or  successively.  In 
other  words  the  mind's  sphere  of  immediate  activity,  unaided  by 
memory,  covers  not  only  the  present  instant  'now,'  but  a  short  reach 
or  length  of  time.  Of  course,  we  should  still  have  to  explain  why  in 
sequence  tones  do  not  fuse  in  the  same  way  as  when  simultaneous. 

This  way  of  accounting  for  the  relations  of  consonance  both  to 
simultaneity  and  succession  of  tones  by  the  same  principles  seems 
easy  only  because  no  definite  theory  has  been  advanced.  Only  the 
formal  requirements  of  a  successful  explanation  have  been  sketched. 
The  other  theories,  such  as  Helmholtz's,  failed  decisively  because  they 
claimed  to  have  found  a  definite  cause  of  consonance  or  dissonance 
which  criticism  has  shown  to  have  apparent  validity  only  in  respect 
of  simultaneous  or  successive  consonance  and  to  be  obviously  inapplicable 
to  the  complementary  case.  And  Stumpf  did  not  feel  quite  satisfied 
with  this  extension  of  his  notion  of  '  synergy ' ;  for  he  suggested  various 


V]  THE  CONSONANCE  OF  SUCCESSIVE  TONES  33 

• 
means  whereby  the  relations  of  successive  tones  might  be  brought 
into  closer  parallel  with  the  fusion  of  simultaneous  ones,  including  the 
principle  of  relationship  through  partials,  advocated  by  Helmholtz 
(cf.  19,  58 fi.).  Coincident  partials  may  well  give  another  kind  of  bond 
between  tones,  but  that  cannot  be  the  bond  of  consonance,  if  consonance 
is  to  be  explained  by  'synergy.'  We  need  not  debate  these  notions 
further  now.  Let  us  rather  consider  the  problem  from  the  volumic 
point  of  view. 

It  is  immediately  clear  that  two  tones  an  octave  apart  in  sequence 
must  have  a  special  relation  to  one  another  as  volumes.  The  higher 
one  will  fall  in  the  tonal  field  exactly  upon  the  upper  half  of  the  volume 
that  formed  the  lower  tone.  Or  the  lower  will  occupy  just  twice  the 
volume  occupied  by  the  previous  higher  tone;  its  pitch-point  will  be 
exactly  where  the  lower  end  of  the  volume  of  the  higher  tone  lay. 
When  the  tones  are  simultaneous,  we  notice  how  perfectly  the  two  fit 
together  to  form  a  regular  whole.  Perhaps  we  get  our  impression  here 
more  as  a  whole  than  from  an  analytic  study  of  the  coincidence  of  points. 
The  coincidence  is  there,  of  course;  but  we  probably  feel  the  fit  as 
a  whole  rather  than  see  it  or  inspect  it  point  for  point.  When  tones 
follow  one  another,  however,  this  analytic  procedure  becomes  more 
possible.  We  could  not,  of  course,  state  in  exact  conceptual  terms  our 
procedure  in  observing  the  tones,  so  as  to  corroborate  precisely  the 
theory  of  their  volumes.  But  the  different  ways  we  use  our  attention 
might  really  correspond  to  the  statements  we  deduce  from  the  theory 
of  the  volumes  of  tones  for  all  that. 

Thus,  e.g.  in  noticing  the  lower  tone  after  a  few  trials  we  may  well 
fixate  the  pitch  of  it  exactly  and  observe  then  whether  the  lower  end 
of  the  higher  tone  just  touches  off  that  pitch-point.  In  vision  we  can 
describe  the  procedure  of  the  attention  in  very  precise  conceptual 
terms.  We  take  one  line  and  let  the  end  point  of  it  fall  exactly  on  the 
end  point  of  another  line  and  make  a  second  point  of  the  line  fall  on 
some  other  point  and  so  on.  We  are  unable  to  do  this  in  hearing,  not 
because  the  stuff  of  sound  would  not  allow  of  it,  nor  because  our  minds 
are  somehow  befogged  in  dealing  with  tone,  but  simply  because  we 
cannot  turn  and  move  tones  about  in  the  auditory  field  as  we  move 
figures  in  the  visual  field.  Nor  can  we  dot  any  required  point  into  a 
sound  volume  as  we  do  with  visual  lines,  and  so  on. 

Similarly  in  the  fifth  we  pass  from  the  one  tone  to  the  other  by  an 
easy  path.   We  could  not  fail  to  notice  the  symmetrical  relation  of  the 

W,  F.  M.  3 


34  THE  CONSONANCE  OF  SUCCESSIVE  TONES  [ch. 

new  defining  points  of  the  higher  volume  to  the  pitch  of  the  lower, 
even  if  it  were  quite  impossible  to  sound  two  tones  at  once. 

But  it  is  obvious  that  there  is  a  considerable  difference  between 
simultaneity  and  succession.  The  former  creates  a  balanced,  unitary 
mass  that  differs  from  other  such  masses  in  its  degree  of  balance  and 
unity.  Sequence  creates  a  passage  that  may  be  regarded  in  much  the 
same  way.  The  lower  tone  gives  way  to  its  octave  gracefully,  as  it 
were;  it  almost  introduces  it,  pointing  in  a  sense  to  the  place  where 
it  will  appear,  or  preparing  a  place  for  it  against  its  coming.  The  same 
is  true  in  a  different  manner  for  the  fifth. 

At  the  same  time  it  is  quite  possible  for  the  mind's  eye  to  take  the 
measure  of  the  two  tones  in  volumic  projection  upon  one  another,  as 
it  were,  and  to  see  their  volumes  against  one  another  as  if  they  were 
simultaneous,  without,  of  course,  being  so.  That  is,  we  can,  if  we  will, 
take  a  sequence  of  tones  as  if  they  were  a  fusion  of  simultaneous  tones 
and  judge  them  accordingly.  The  two  tones  do  not,  of  course,  actually 
fuse ;  but  they  have  to  be  taken  or  heard  together  as  if  the  first  one  were 
still  there  when  the  second  appears ;  their  intensities  are  not  summated 
as  in  the  case  of  simultaneous  fusion  (cf.  p.  52  f.  below),  but  the  balance 
and  symmetry  of  their  relative  positions  are  noted.  This  is  done 
regularly  in  music  in  the  arpeggio  forms  of  chords.  But  it  is  not 
necessary  on  the  other  hand  that  the  mind  should  always  do  so.  There 
is  not  only  no  reason  why  it  should,  but  it  is  easy  for  it  to  do  otherwise. 

Special  interests  of  music,  especially .  the  melodic,  lead  us  to  take 

successive  tones  specifically  as  a  sequence.    Here,  on  the  contrary, 

the  tones  are  apprehended  specially  as  a  sequence;  we  let  the  first 

one  go  and  pass  from  it  to  the  second.   In  the  arpeggio  chord  we  have 

a  whole  given  successively;  in  the  melody  a  sequence  or  motion  is  given 

successively.   Or  in  the  chord  the  successive  tones  are  held  in  projection 

upon  one  another,  while  in  the  melody  they  are  each  complete  stages 

of  a  transition.    For  this  purpose  smallness  of  interval  is  a  favouring 

factor;  it  makes  for  continuity  or  for  melodic  progression^.    Continuity 

is  present  even  with  the  larger  interval,  but  it  is  then  not  so  obvious 

or  so  obtrusive;  it  may  have  to  be  supported  by  other  relations  which 

bring  successive  tones  into  connexion  with  one  another.    In  this  way 

*  Cf.  41,  346:  "In  folk  music  generally  the  frequency  with  which  the  various  intervals 
are  used  decreases  proportionately  with  their  size."  It  does  so  also  in  the  melodies  of 
Schubert's  songs,  as  I  have  determined  by  sampling  every  tenth  song.  Only  the  minor 
second  occurs  less  frequently  than  the  major.  The  figures  of  the  several  frequencies  are : 
2—1673,  11—2171,  3—926,  III-466,  4—633,  T— 60,  5—195,  6—118,  VI— 58,  7—17, 
VII— 0,  0—28,  9—1,  IX— 0,  10—1,  X— 2.    The  sample  consisted  of  56  songs. 


V]  THE  CONSONANCE  OF  SUCCESSIVE  TONES  35 

we  very  often  find  the  melodic  and  the  consonantal  aspects  conjoined  in 
the  same  interval.  Large  intervals  enter  into  melodies  more  easily  when 
they  are  such  as  would  be  consonances  with  simultaneous  tones  (cf. 
52,  22)1.  j^Q^  ^j^a^  ^Y^Q  greatest  consonance — the  octave — is  oftenest 
used,  the  fifth  next,  and  so  on.  Each  interval  has  to  be  judged  on  its 
merits.  The  great  consonance  of  the  octave  is  weighed  down  by  the 
large  melodic  step  required  by  it  and  is  probably  less  often  used  for 
that  reason.  The  fifth  with  a  lesser  consonance  will  very  likely  be  used 
oftener  because  of  the  greater  advantage  given  by  its  much  smaller 
step.  The  matter  has  not  been  fully  treated  statistically,  as  far  as  I 
am  aware,  but  it  would  probably  well  repay  the  trouble  necessary  to 
gather  the  facts. 

Thus  it  seems  that  in  the  volumic  theory  a  basis  is  presented  from 
which  all  the  interests  of  music  in  simultaneous  and  successive  tones 
may  be  fully  satisfied  without  neglect  of  any  of  the  differences  involved 
in  these  two  cases-. 

^  This  is  confirmed  in  the  statistics  of  Schubert's  songs. 


3—2 


CHAPTER  VI 

THE  NATURE  OF  INTERVAL 

The  nature  of  interval  has  always  been  one  of  the  great  mysteries  of 
sound.  It  formed  for  the  ancient  world  a  fitting  parallel  in  sense  to 
the  wonderful  relations  shown  by  numbers.  The  discovery  of  the 
connexion  between  the  grades  of  consonance  and  the  ratios  of  the 
smaller  numbers  let  loose  a  flood  of  mysticism  which  endured  for 
centuries.  Rameau  seems  even  to  have  thought  that  a  thorough 
explanation  of  the  sensory  basis  of  tonal  proportion  might  lead  to  an 
insight  into  the  being  of  proportion  in  general  and  in  particular  as  it 
appears  to  us  in  numbers^.  He  was  sharply  criticised  for  this  by  the 
Academic  des  Sciences  to  whom  he  presented  his  scientific  plans  for 
approval  and  support.  And  their  censure  of  his  mystical  vanities  was 
re-voiced  by  D'Alembert^  in  spite  of  the  admiration  which  Rameau's 
efforts  and  success  in  forming  a  systematic  whole  out  of  the  empirical 
musical  wisdom  of  his  time  aroused  in  him.  Rameau,  of  course,  did 
not  succeed  in  explaining  the  mystery  of  interval  and  its  relation  to 

^  53,  2:  "Ne  connoissant  point  la  nature  de  notre  Ame,  nous  ne  pouvons  appretier 
les  rapports  qui  se  trouvent  entre  les  differens  sentimens  dont  nous  sommes  affectes: 
cependant  lorsqu'il  s'agit  des  Sons,  nous  supposons  qu'ils  ont  entr'eux  les  memes  rapports 
qu'ont  entr'elles  les  causes  qui  les  produisent."  "Ce  qu'on  a  dit  des  Corps  sonores  doit 
s'entendre  egalement  des  Fibres  qui  tapissent  le  fond  de  la  Conque  de  1' Oreille;  ces  Fibres 
sont  autant  de  corps  sonores  auxquels  I'Air  transmet  ses  vibrations,  et  d'oA  le  sentiment 
des  Sons  et  de  THarmonie  est  porte  jusqu'i  I'Ame"  (p.  7).  "On  peut  dire  meme  que  la 
Musique  a  cet  avantage  singulier,  qu'elle  peut  toujours  offrir  en  meme-tems  k  I'esprit  et 
aux  sens  tous  les  rapports  possibles  par  le  molen  d'un  Corps  sonore  mis  en  mouvement; 
au  lieu  que  dans  les  autres  parties  des  Mathematiques  I'esprit  n'est  pas  ordinairement  aide 
par  les  sens  pour  appercevoir  ces  rapports"  (Epitre). 

2  "Le  corps  sonore  ne  nous  donne  et  ne  peut  nous  donner  par  lui-meme  aucune  idee 
des  proportions.. .  .3°.  (et  c'estici  la  raison  principale)  parce  que,  quand  on  entendrait  ces 
octaves  et  ces  sons  des  multiples,  le  sens  de  I'ouie  ne  peut  en  aucune  maniere  nous  donner 
la  notion  de  rapport  et  de  proportion,  que  nous  ne  pouvons  acquerir  que  par  la  vue,  et 
par  le  toucher.  Pour  avoir  une  idee  nette  des  proportions  et  des  rapports,  il  est  necessaire 
de  comparer  les  corps  par  ces  deux  demiers  sens;  la  perception  des  sons  n'y  contribue 
absolument  en  rien,  n'y  ajoute  rien,  y  est  totalement  ^trangere.  Pour  tout  dire  en  un 
mot,  quand  les  hommes  seraient  sourds,  il  n'y  en  aurait  pas  moius  pour  eux,  des  rapports, 
des  proportions,  une  geometric.  En  voila,  Monsieur  [Rameau],  plus  qu'il  n'en  faut  sur 
ce  sujet;  et  les  Mathematiciens  trouveront  k  coup  sur  que  j'en  ai  encore  trop  dit"  (9, 2i3f.). 


CH.  VI]  THE  NATURE  OF  INTERVAL  37 

physical  ratios.  But  he  was  certainly  right  in  feeling  that  there  was 
something  in  the  sensory  experience  to  be  explained  which  would  tell 
us  how  we  feel  proportion  in  one  instance  at  least,  whether  that  case 
can  throw  any  light  upon  other  forms  of  mentally  grasped  proportion 
or  not.  He  did  well  to  linger  longingly  upon  the  wondrous  problem. 
And  D'Alembert's  denial  was  somewhat  too  sweeping,  at  least  so  far 
as  the  presence  of  proportion  in  hearing  is  concerned. 

That  we  detect  very  special  and  precise  features  in  our  tonal 
experiences  correlated  to  certain  very  definite  proportions  in  their 
physical  stimulus,  should  prevent  a  cautious,  logical  mind  from  asserting 
point  blank  that  proportion  has  absolutely  no  place  in  hearing. 

D'Alembert,  like  so  many  others  since  his  day,  was  convinced  that 
the  'metaphysics'  or  psychology  of  hearing  would  "according  to  all 
appearances  always  remain  covered  with  clouds."  And  yet  he  somehow 
convinced  himself  at  the  instigation  of  Rameau's  researches  that  the 
principal  laws  of  harmony  could  be  deduced  from  a  single  experiment 
(9,  xxvii).  But  we  now  know  that  that  idea  is  really  as  absurd  as 
Rameau's  speculations  on  proportion  seemed  to  D'Alembert  himself. 
In  fact  it  is  worse.  For  Rameau  did  include  the  phenomena  of  hearing 
in  his  field  of  search,  whereas  D'Alembert  seems  to  have  thought  that 
a  physical  experiment  or  relation  was  worthy  of  the  place  of  honour 
at  the  feast  of  music  without  wearing  the  garment  of  experience. 
Criticism  has  since  thrown  that  and  all  other  intruders  out  into  the 
limbo  they  belong  to.   And  the  clouds  have  blown  away. 

In  fact  the  solution  is  not  by  any  means  difficult  to  attain  or  to 
apprehend,  once  the  fundamental  secret  of  tone  has  been  discovered 
and  understood.  However  that  may  be,  the  wonder  of  it  all  remains 
that  sound  and  hearing  should  be  so  cunningly  devised;  that  the  weft 
of  nature's  mighty  looms  should  reveal  so  beautiful  a  pattern  in  this 
auditory  part.  And  the  greater  wonder  still  is  that  our  intellect  should 
have  been  able  from  this  slender  basis  to  raise  the  great  art  of  music 
to  such  complexity.  Our  task  in  the  following  pages  will  be  to  try  to 
show  how  the  great  edifice  of  music  is  placed  secure  on  it«  foundations 
and  how  it  is  carried  upwards  towards  the  art  as  we  know  it.  No  one 
who  follows  this  science  of  tone  from  its  beginnings  can  fail  to  be  struck 
by  the  extraordinary  nature  of  sound  and  the  marvellous  skill  with 
which  music  has  been  created  by  man. 

The  oldest  theory  from  which  an  explanation  of  interval  was  sought 
started  from  the  obvious  fact  of  vibration  in  the  sonorous  body  and  the 


38  THE  NATURE  OF  INTERVAL  [ch. 

relation  between  pitch  and  rate  of  vibration.  "This  doctrine,  first 
taught  by  the  illustrious  founder  of  the  sect  [Pythagoras],  adopted 
and  developed  by  Lasos,  by  Aristotle,  by  Euclid,  and  later  by  the 
neo-platonists,  has  been  formulated  by  Nicomachus,  whose  words 
Boethius  transmits  to  us.  'It  is  not,'  he  says,  'a  vsingle  vibration 
that  produces  a  uniform  sound;  but  the  string,  once  set  in  motion, 
gives  birth  to  numerous  sounds,  because  it  impresses  frequent  vibrations 
upon  the  air.  But,  as  the  rapidity  of  these  shocks  [of  the  air]  is  so  great 
that  one  sound  is  confounded  in  some  way  with  the  other,  we  do  not 
perceive  the  distance  [that  separates  them],  and  it  is  as  it  were  a  single 
sound  that  reaches  our  ears.  Now  when  the  vibrations  of  the  low  notes 
and  the  high  notes  are  commensurable  amongst  themselves  (as  for 
example  in  the  proportions  indicated  above),  there  is  no  doubt  that 
these  common  measures  blend  together  and  produce  the  unity  of  sounds 
we  call  consonance'"  (14,  96 f.;  3,  I,  3i)i.  Keeping  in  touch  with  the 
progress  of  the  physical  science  of  sound,  this  doctrine  has  been  carried 
down  to  our  own  time.  It  was  the  basis  of  Euler's  Tentamen  novae 
iheoriae  musicae  ex  certissimis  harmoniae  princi'piis,  from  which  the 
Table  opposite  his  page  36  has  been  copied  so  often.  Thus  the  octave 
gives  a  pattern  of  this  kind — :  •  :  •  :  •  : — the  upper  line  of  dots 
representing  the  waves  of  the  higher  tone,  the  lower  the  slower  waves 
of  the  lower  tone.  The  pattern  for  the  fifth  would  be  thus — :  ■.*  :  '."  : 
(2  :  3).  Probably  the  best  and  at  the  same  time  the  most  self-critical 
statement  this  theory  has  ever  received  was  made  by  a  Scotsman,  John 
Holden,  in  an  Essay  towards  a  rational  system  of  music,  published  in 
Glasgow  in  1770.  The  psychological  analogies  he  brought  forward  are 
admirable.  Even  in  recent  years  the  theory  has  been  renovated  by 
Th.  Lipps,  who  believed  that  these  waves  were  carried  to  the  brain 
and  transferred  to  subconsciousness,  there  being  a  unit  of  process  in 
the  latter  for  each  physical  wave.  Somehow  this  sequence  took  on  for, 
or  in,  consciousness  the  form  of  a  smooth  unity.  The  rhythmic  coin- 
cidence of  the  processes  of  subconsciousness  that  went  on  when  two 
tones  were  sounded,  was  supposed  to  be  felt  by  consciousness  as  con- 
sonance, want  of  rhythm  or  its  puzzling  complexity  as  dissonance. 

^  "Non,  inquit,  unus  tantum  pulsus  est,  qui  simplicem  modutn  vocis  emittat,  sed 
semel  percussus  nervus  saepius  aerem  pellens  multas  efficit  voces.  Sed  quia  ea  velocitas 
est  percussionis  ut  sonus  sonum  quodammodo  comprehendat  distantia  non  sentitur,  et 
quasi  una  vox  auribus  venit.  Si  igitur  percussiones  gravium  sonorum  commensurabiles 
sint  percussionibus  acutorum  sonorum,  ut  in  his  proportionibus  quas  supr^  retulimus, 
non  est  dubium  quin  ipsa  commensuratio  sibimet  misceatur,  unamque  vocum  efficiat 
consonantiam." 


VI]  THE  NATURE  OF  INTERVAL  39 

"It  is  not,"  as  John  Curwen  said  (8,  lo),  "that  the  mind  actually  cowwte 
the  relative  number  of  vibrations  and  consciously  ascertains  that  one 
tone  gives  exactly  half  as  many  as  the  other.  But  by  one  of  those 
rapid  though  complex  mental  processes  which  are  the  marvel  of  the 
philosopher,  it  feels  the  result,'"  adding  afterwards  in  a  similar  context, 
"in  a  way  the  Great  Creator  only  knows." 

The  earUest  forms  of  this  theory  were  a  legitimate  attempt  to  bring 
into  connexion  the  two  ends  of  the  psycho-physical  process,  where 
knowledge  offers  itself  most  readily, — the  physical  and  the  auditory. 
But  for  later  theorists,  who  realise  the  gap  there  is  for  all  systematic 
possibilities  between  merely  felt  grades  of  consonance  and  ratios  of 
physical  vibration,  whether  of  the  air  or  of  the  ear,  the  theory  is  merely 
an  effort  to  make  bricks  of  straw.  Besides,  as  Rousseau  noticed  (56, 
Art.  'Consonance,'  14th  paragraph),  how  is  the  mind  to  catch  the 
rhythm  or  whatever  it  may  be  called,  when  the  periods  do  not  begin 
and  end  at  the  same  time,  or,  as  we  now  say,  when  their  phases  are 
not  properly  coincident?  We  shall  not  spend  time  discussing  any 
forms  of  the  theory.  There  is  nothing  to  discuss  but  mere  speculation 
or  ignorance  trying  to  "materialise"  itself  to  knowledge.  Ignorance 
does  not  breed  knowledge ;  it  is  the  waste  land  of  science  to  be  gradually 
conquered  by  the  shoots  of  knowledge  that  spread  into  it.  Every 
attempt  to  bridge  the  gap  between  vibration  and  sound  must  rest 
upon  greater  success  in  the  description  of  the  physical  process  or  of  the 
sounds  themselves.  For  a  complete  study  of  either  must  finally  lead 
to  the  other,  just  as  one  real  process  binds  the  two  into  a  single  event. 
In  this  case  our  way  of  success  begins  from  the  auditory  side. 

The  line  of  progress  is,  in  fact,  continuous  with  the  theory  of  tones 
already  developed.  It  is  easily  seen  that  if  the  upper  tone  fits  so  perfectly 
into  the  lower  tone  in  the  case  of  the  octave,  it  will  do  so,  however 
large  the  volume  of  the  lower  tone  may  be,  so  long  as  its  volume  is  the 
perfect  fit.  The  lower  tone  may  be  moved  gradually  from  the  lowest 
reach  of  the  musical  range  of  pitch  till  the  upper  tone  reaches  the 
opposite  extreme.  What  is  common  in  this  series  will  constitute  the 
interval  of  the  octave  as  against  its  fusion  or  consonance.  What  is  this 
common  feature? 

At  first  glance  there  seems  to  be  nothing  that  one  can  claim  as  the 
basis  of  interval,  since  the  balance  or  unity  of  the  whole  has  been 
allocated  to  the  heard  fusion.  Even  if  this  allocation  was  in  the  first 
instance  the  outcome  of  a  process  of  logical  exclusion  (77,  60 ff.)  it  is 


40  THE  NATURE  OF  INTERVAL  [ch. 

confirmed  by  the  kinship  of  the  two  terms  thus  brought  into  connexion 
— namely  heard  fusional  unity  or  balance  and  conceptually  formulated 
balance  or  unity.  A  further  process  of  discovery  by  exclusion  seems 
difl&cult  in  the  case  of  the  octave,  because  the  fusional  aspect  of  the 
bi-tonal  mass  is  here  so  prominent,  both  for  sense  and  for  conception. 
Let  us  therefore  consider  a  case  from  the  lower  grades  of  fusion. 

There  is  only  a  slight  difference  between  the  major  and  the  minor 
thirds  or  between  the  different  seconds  and  sevenths  in  the  matter  of 
consonance  or  dissonance.  If  these  bi-tonal  masses  had  no  other  feature 
than  their  fusion  they  would  never  have  become  so  distinctive  as  they 
now  are  in  music.  Then  there  must  be  some  other  feature  in  them  that 
provides  a  basis  for  our  sense  of  interval. 

Let  us  abstract  for  a  moment  from  balance  and  unity  altogether,  as 
if  we  did  not  apprehend  it.  Then  we  may  make  the  following  assertion. 
So  long  as  we  were  capable  of  noting  the  pitches  and  the  volumes  of 
tones,  even  supposing  they  did  not  overlap  (provided  only  they  consisted 
of  a  number  of  particles  or  'spots'  of  sound,  each  ordinally  distinct 
and  fixed,  and  so  capable  of  being  repeated  precisely  any  number  of 
times),  we  should  still  be  able  to  note  the  relative  proportions  of  their 
volumes  and  to  construct  to  any  given  volume  X  a  volume  Y  so  that 
their  proportion  should  be  the  same  as  that  of  a  standard  pair  P  and  Q. 
We  might  not  be  able  to  do  this  as  well  as  we  judge  and  reproduce 
intervals  under  present  circumstances.  Our  margin  of  error  would 
probably  be  greater,  just  as  it  is  when  we  compare  the  lengths  or  pro- 
portions of  visual  lines  from  an  unusual  standpoint.  We  usually  place 
them  directly  in  front  of  us  and  squarely  to  the  line  of  sight.  If  we 
could  judge  the  proportions  of  volumes  under  these  circumstances,  it 
must  be  evident  that  the  comparison  of  the  relative  volumes  of  a  pair 
of  tones  is  not  made  more  difl&cult  by  the  fact  that  the  lower  volume 
consists  partly  of  or  includes  the  volume  of  the  higher  tone,  so  long 
as  the  higher  volume  is  distinguishable  in  the  lower.  In  fact  it  may 
well  be  easier;  for  the  volumes  appear  in  the  same  place  in  the  auditory 
field.  And  the  ease  of  comparison  is  increased  by  the  facility  with  which 
the  volumes  can  be  observed  in  succession. 

In  thus  appealing  to  a  sense  of  proportion  we  are  merely  giving 
greater  scope  to  a  faculty  of  mind  that  experimental  study  has  in  recent 
years  shown  to  be  of  the  greatest  importance  and  of  the  finest  efl&ciency. 
If  a  visual  standard  of  proportion  is  given,  say  two  lines  forming  the 
sides  of  a  parallelogram,  a  fourth  line  can  be  constructed  to  a  given 
third  that  will  show  the  same  proportion  with  only  a  very  slight  error. 


VI]  THE  NATURE  OF  INTERVAL  41 

The  same  sort  of  proportion  can  be  carried  through  even  with  intervals 
of  time. 

In  any  case  since  tonal  intervals  can,  as  a  matter  of  fact,  be  so  finely 
learnt  and  reproduced  as  every  musician  knows,  and  since  there  is  so 
strong  evidence  that  tones  are  volumes  of  sound  of  definite  magnitudes, 
consisting  of  a  definite  part  of  a  fixed  series  of  auditory  particles,  each 
differing  from  the  other  only  in  its  place  in  the  series,  we  have  every 
right  to  claim  that  the  real  basis  of  our  sense  of  interval  is  our  observation 
of  a  constant  proportion  between  the  volumes  of  tones. 

This  claim,  though  it  has  been  won  by  careful  theory,  that  carries 
our  minds  through  and  beyond  what  the  bare  tones  themselves  suggest 
to  our  simple  observation,  is  in  the  end  confirmed  by  our  observation. 
We  have  only  to  ask  ourselves  :  is  not  what  we  call  interval  a  constant 
proportion  between  tones  as  we  hear  them?  We  shall  perhaps  not  assent 
at  once  if  we  merely  observe  a  single  interval  reflectively.  But  take 
that  interval  and  think  it  successively  on  to  a  long  series  of  tones  of 
different  pitch.  Such  a  test  will  show  that  we  are  in  every  way  as  fully 
justified  in  translating  sense  of  interval  into  sense  of  proportion  as  we 
are  in  speaking  of  a  sense  of  proportion  in  any  department  of  experience 
at  all.  We  are  in  the  sense  of  interval  finding  the  proportions  of  things 
that  really  bear  proportion  to  one  another,  and  we  do  so  very  accurately. 

When  we  establish  relations  of  proportion  between  lengths  of  line 
by  mere  visual  inspection  of  them,  what  do  we  do?  And  what  are  we 
aware  of?  We  inspect  these  lines  and  compare  them  as  to  their  lengths 
which  appear  as  sensible  magnitudes.  We  base  our  judgment  of  propor- 
tion upon  this  inspection.  We  are  aware  of  the  magnitudes  we  inspect 
as  lengths  and  we  feel  keenly  whether  a  known  or  given  standard  of 
proportion  is  repeated  in  a  given  pair  of  lines,  making  thereby  in  our 
judgments  only  a  very  small  margin  of  error.  In  judging  the  proportions 
of  tones,  or  in  judging  tonal  interval  we  do  exactly  the  same.  We  inspect 
tonal  lines  of  a  little  breadth,  or,  as  we  usually  call  them,  tonal  volumes. 
In  these  volumes  pitches  appear,  not  detracting  from  our  power  to 
judge  of  interval,  but  rather  aiding  it  considerably  by  giving  it  a  sort 
of  focus.  We  are  aware  that  the  tones  we  compare  have  volumes  or 
that  the  whole  volume  constituting  an  interval  has  a  particular  volumic 
figure.  We  are  aware  of  this  even  though  we  could  not  describe  what 
we  are  aware  of  in  the  clear  conceptual  terms  we  readily  use  in  vision. 
For  in  vision  we  are  all  both  in  practice  and  in  theory  highly  expert, 
whereas  in  hearing  most  of  us  are  in  practice  very  inexpert  and  we  have 
all  been  devoid  of  proper  theoretical  insight.   Now  that  the  insight  has 


42  THE  NATURE  OF  INTERVAL  [ch. 

come,  we  can  see  that  it  gives  a  true  description  of  what  we  do  and  of 
what  occupies  our  attention  while  we  estimate  interval.  In  judging 
interval  we  also  feel  keenly  whether  a  known  or  given  standard  of 
proportion  is  repeated  in  a  given  interval  and  the  margin  of  error  made 
by  expert  judges  is  very  small. 

The  study  of  these  lower  grades  of  consonance  as  intervals  shows 
us,  moreover,  that  we  can  fix  any  interval  as  an  interval  in  our  memory. 
The  interval  may  be  24  :  31  equally  as  well  as  24  :  32,  provided  it  be 
fixed  in  the  memory  by  frequent  repetition  and  attention.  Of  course 
it  is  much  easier  to  learn  the  consonant  intervals  because  they  have  a 
special  attraction  for  the  attention  and  for  the  memory.  For  on  the 
one  hand  they  fuse,  and  on  the  other  they  are  few  and  important. 
Intervals,  such  as  the  tritone  and  the  major  seventh,  which  differ  only 
a  little  in  size  from  some  prominent  consonance,  are  hard  to  sing  because 
they  tend  to  slide  into  the  easy  consonance,  as  it  were.  But  with  sufficient 
practice  any  such  difficulty  may  be  overcome.  As  Alfred  Day  said  in  a 
similar  connexion  :  "Practice  is  for  the  purpose  of  overcoming  difficulties 
and  not  of  evading  them"  (10,  7). 

It  is  conceivable,  as  some  have  claimed,  that  those  who  constantly 
practise  with  the  intervals  of  equal  temperament  should  finally  come 
to  ase  them  and  to  think  in  them  by  preference^.  When  the  circumstances 
of  judging  are  most  favourable,  the  accuracy  with  which  deviations 
from  a  familiar  interval  can  be  detected  is  very  great.  Thus  Meyer 
and  Stumpf  got  collective  results  showing  inter  alia  an  accuracy  of 
74  %  for  a  deviation  of  —  0*78  vb.  from  an  ascending  major  third 
(600  vbs.);  +2-18  gave  72%.  An  individual  result  (Stumpf's)  gave 
88  %  for  -  0-78  from  the  ascending  third  and  82  %  for  +  2-18  (73, 

358  ff.). 

This  process  of  abstraction  has  thus  yielded  us  a  new  feature  of 
complex  volumes,  namely  the  proportion  of  their  parts  or  interval. 
We  may  now  look  back  upon  the  well-balanced  fusions  that  seemed  to 
offer  no  other  feature  for  analysis  than  their  obvious  balance,  and 
reconsider  the  problem. 

The  octave,  we  may  say  now,  is  not  only  a  special  fusion;  it  is  an 
interval  as  well.  The  tones  that  form  it  do  not  only  fit  peculiarly  into 
one  another,  but  they  also  bear  a  certain  volumic  proportion  to  one 
another.    Thus  we  have  a  double  basis  by  which  to  fix  the  octave  in 

^  This  sentence  bears  no  reference  to  the  controversy  on  the  respective  merits  of 
equal  and  just  temperament. 


VI]  THE  NATURE  OF  INTERVAL  43 

the  attention  and  memory  and  a  double  use  for  it  in  music.  If  we  ask 
what  are  the  respective  contributions  of  fusion  and  of  interval  to  the 
importance  of  the  role  played  by  the  octave  in  music,  there  can  be 
no  hesitation  as  to  the  answer.  By  far  the  more  important  aspect  of 
it  is  its  fusion.  Had  we  not  a  linear  field  of  sound,  but  an  areal  one, 
as  in  vision,  in  which  tones  could  be  given  at  varying  distances  from 
one  another  without  overlapping  at  all,  we  should  have  attached  as 
little  importance  in  music  to  a  1  :  2  proportion  between  tonal  volumes 
as  we  attach  to  that  proportion  between  the  lengths  of  lines  in  visual 
art.  The  1  :  2  proportion  stands  forth  in  mu&ic  because  the  '  upper ' 
ends  of  all  tones  are  identical  and  tones  overlap  completely  from  thence 
'downwards.'  It  is  idle  to  speculate  as  to  what  kind  of  music  we  might 
have  made  if  we  had  had  such  an  areal,  or  even  a  cubic,  field  of  sound. 
I  mean,  of  course,  areal  for  musical  purposes.  It  is  areal  already,  as 
shown  above,  as  a  whole,  but  the  transverse  dimension  has  no  musical 
utility. 

Interval  might  well  be  called  our  sense  of  form  in  sound,  when 
fusion  would  be  our  sense  of  mass,  as  it  were.  There  is  no  use  in  labouring 
these  analogies  between  sight  and  sound,  except  in  so  far  as  they  help 
to  bring  out  the  underlying  identity  of  structure  in  the  two  senses, 
and  so  to  understand  the  nature  of  each  better.  Still  less  should  we 
attempt  to  base  practical  reforms  or  advances  upon  these  interpretations 
by  trying  to  raise  upon  the  foundations  of  tonal  mass  or  tonal  form 
structures  analogous  to  those  developed  in  the  visual  arts  upon  the 
foundations  of  those  names.  If  such  structures  are  naturally  possible 
to  music,  they  will  probably  have  been  created  to  some  extent  already. 
If  the  analogies  suggested  are  real,  the  first  event  to  follow  may  well 
be  the  discovery  that  certain  types  of  music  differ  by  their  emphasis 
upon  fusion  or  upon  interval,  upon  mass  or  upon  form. 

Possibly  that  is  the  real  meaning  of  the  great  difference  of  nature 
and  view  between  harmonic  and  polyphonic  music,  the  former  being 
the  art  of  mass  of  fusional  (consonantal)  effects,  of  course.  No  visual 
art  is  purely  a  construction  of  masses  or  of  forms  alone.  It  is  impossible 
to  separate  mass  and  form  in  this  way.  Every  mass  must  have  a  form, 
and  every  form  that  is  at  least  bi-dimensional  indicates  mass  to  some 
degree  or  other.  Visual  arts  do,  nevertheless,  differ  in  the  relative  extent 
to  which  they  build  on  mass  and  form.  Similarly  in  music.  Every 
fusion  has  a  form  and  every  interval  has  some  degree  of  balance  or  mass 
unity  about  it.  But  polyphonic  is  commonly  said  to  differ  from  harmonic 


44  THE  NATURE  OF  INTERVAL        [ch.  vi 

music  in  that  the  one  is  viewed  horizontally,  the  other  perpendicularly; 
or  in  that  the  one  regards  chords  rather  as  a  whole,  while  the  other 
takes  more  interest  in  creating  and  following  out  the  lines,  as  it  were, 
that  run  side  by  side  throughout  the  successive  groups  of  sounds.  The 
difference  is  one  of  degree.  We  shall  see  as  we  proceed,  that  this  dis- 
tinction of  attitudes  towards  groups  of  tones  is  of  the  greatest  importance 
for  a  study  of  the  foundations  of  music. 


CHAPTER  VII 

THE  MUSICAL  RANGE  OF  PITCH 

One  of  the  most  curious  fact«  of  hearing  is  that  music  is  restricted  to 
a  certain  range  of  pitch.  Outside  the  limits  thereof  it  is  no  longer 
possible  to  make  music.  The  pianoforte  makes  these  limits  familiar 
to  every  one.  The  lowest  tone  on  the  large  concert  grand  piano  is  A^, 
the  highest  is  c®.  These  pitches  include  a  little  more  than  seven  octaves. 
Anyone  may  notice  upon  the  piano  how  the  lowest  notes  seem  to  give 
an  insufficient  difEerence  of  pitch  from  their  neighbours.  The  intervals 
of  a  major  second  seem  too  small;  those  of  a  minor  second  seem  to  be 
hardly  distinguishable  as  intervals.  A  little  more,  one  thinks,  and  the 
two  tones  would  seem  to  be  the  same.  At  the  upper  end  of  the  keyboard 
a  similar  change  is  noticed,  though  it  is  not  nearly  so  distinct  upon  the 
piano.  But  it  appears  clearly  if  we  carry  the  pitch  of  tone  physically 
some  distance  into  the  c^  octave. 

This  limitation  of  range  does  not  depend  upon  any  purely  physical 
restriction.  Periodic  waves  can  be  produced  below  and  above  these 
limits  and  pairs  of  tones  maintain  their  proper  physical  relations  to 
one  another  unchanged.  Nor  does  the  phenomenon  seem  to  depend 
upon  an  incapacity  of  the  ear  to  hear  tone.  For  the  ear  responds  with 
a  tone-like  sound  to  physical  rates  of  vibration  at  least  four  or  five 
times  as  great  as  that  required  for  c*.  A  great  deal  of  patient  effort 
and  ingenuity  has  been  spent  upon  the  attempt  to  fix  the  vibrational 
limits  of  hearing  as  accurately  as  possible.  They  will  always  be  uncertain ; 
for  their  physical  sources  are  not  only  hard  to  control  and  to  gauge 
correctly,  but  the  range  of  hearing  varies  considerably  from  person 
to  person  and  from  youth  to  age.  It  also  varies  considerably  with  the 
intensity  of  the  physical  stimulus.  In  fact  it  is  possible  that  within 
a  large  range  of  physical  differences  any  rate  of  vibration  will  produce 
some  auditory  effect  or  other,  and  if  loud  enough  it  may  be  a  tonal 
effect,  without  there  being  any  real  differences  in  these  effects,  except 
minor  or  accidental  ones.  The  determination  of  the  limits  of  hearing 
would  thus  be  illusory,  after  a  certain  point.  We  shall  realise  this 
possibility  better  further  on. 

Material  has  been  gathered  carefully  by  experimental  means  towards 


46  THE  MUSICAL  RANGE  OF  PITCH  [ch. 

an  adequate  description  of  the  limits  of  the  musical  range  of  tone.  It  is 
found  that  no  sharp  boundaries  mark  it  out.  Towards  the  upper  side 
it  shows  itself  first  in  a  slight  apparent  flattening  of  the  pitch  of  tones 
from  what  the  rate  of  vibration  of  its  source  leads  us  to  expect. 
Gradually  as  the  tone  is  raised  this  flattening  increases  to  a  semitone, 
and  even  to  a  tone.  Beyond  this  point  the  estimation  of  pitch  soon 
breaks  down  altogether.  A  similar  gradual  deterioration  of  judgment 
is  found  on  the  lower  side,  but  here  the  very  low  tones  seem  to  be 
sharper  than  they  should  be,  according  to  the  known  physical  rates 
of  vibration. 

If  interval  is  constituted  by  constant  proportion  of  volume,  it  follows 
that  so  long  as  the  pitch  of  a  tone  of  these  high  or  low  regions  can  be 
estimated  with  confidence  and  regularity,  there  is  at  least  on  the 
phenomenal  side  or  in  the  tones  themselves  nothing  amiss.  The  octave 
is  still  in  every  way  an  octave  for  hearing.  The  discrepancy  lies  only 
between  the  auditory  and  the  physical  series.  It  requires  a  greater 
ratio  of  physical  vibration  to  produce  a  volumic  octave  in  the  high 
border  region  than  is  usual  in  the  middle  range  of  hearing.  The  ratio 
is  1:2+,  instead  of  1:2.  Similarly  in  the  low  border  region  the 
'physical  ratio  downwards  is  1  :  |  —  instead  of  exactly  1  :  |.  On  the 
higher  side  there  is  therefore  evidently  some  difficulty  in  making  the 
volume  of  tone  smaller  in  the  usual  proportion.  So  the  rate  of  vibration 
has  to  be  increased  a  little  in  order  to  get  a  reduction  of  the  volume 
by  half  exactly.  On  the  lower  side  there  is  evidently  a  difficulty  in 
making  a  volume  of  the  usual  large  size.  The  sizes  required  are  so  great 
that  a  reduction  of  the  rate  of  vibration  beyond  the  half  is  necessary 
in  order  to  get  precisely  the  double  volume. 

These  special  difficulties  receive  an  easy  explanation  by  reference 
to  the  physical  sense-organ.  The  cochlea  is  quite  a  small  thing,  and 
although  its  functions  seem  to  be  remarkably  independent  of  its  size, 
they  can  be  so  only  relatively,  not  absolutely.  There  must  come  a  point 
at  which  the  organ  will  fail  to  respond  properly  to  a  very  short  wave- 
length of  vibration.  Similarly  it  will  be  at  some  point  or  other  finally 
unable  to  accommodate  the  great  long  waves  of  sound.  No  apparatus 
at  all  will  cover  an  infinite  range  of  forces.  It  will  fail  to  work  beyond 
certain  extremes,  and  towards  these  it  will  lag  behind  the  change  of 
force  applied.  At  first  this  will  be  only  a  perceptible  lag,  finally  no 
further  change  will  be  given.  The  ear  will  respond  with  the  extreme 
possible  to  it  on  either  side. 

This  seems  perhaps  to  be  the  case  at  the  upper  pitch  limit  of  hearing. 


VII]  THE  MUSICAL  RANGE  OF  PITCH  47 

For  a  considerable  period  beyond  the  end  of  the  musical  range  tones 
are  still  heard.  They  become  slowly  thinner  and  sharper  and  finally 
disappear  gradually  into  a  mere  hiss  or  puff.  At  the  lower  end  the  longer 
waves  seem  after  a  time  to  produce  no  more  effect  upon  the  ear  at  all. 
At  the  most  they  are  felt  as  puffs  of  air  against  the  drum  of  the  ear. 

The  musical  range  of  hearing,  then,  is  the  range  within  which  the 
change  of  tonal  volume  keeps  march  with  the  change  of  vibratory 
rate.  As  long  as  this  holds  good,  instruments  may  be  constructed  and 
played  with  freedom  and  with  complete  certainty  as  to  the  musical 
effect  upon  the  listener.  It  is,  of  course,  conceivable  that  musical 
work  could  be  carried  up  to  the  outer  limits  of  the  border  region  for 
a  single  listener  at  least.  But  the  physical  ratios  required  at  these 
last  points  in  order  to  maintain  the  desired  relationships  of  tonal  volume 
would  not  be  generally  valid.  They  vary  considerably  from  person  to 
person.  Consequently  music  is  more  or  less  obliged  to  discard  these 
border  regions  in  so  far  as  precise  effects  are  desired.  Thus  the  musical 
range  of  tone  becomes  the  range  within  which  the  changes  of  volume 
and  of  vibratory  rate  are  exactly  inversely  proportional  to  one  another. 


CHAPTER  VIII 

OUR  POINT  OF  VIEW  TOWARDS  THE  AUDITORY  FIELD 

When  the  physics  and  physiology  of  vision  had  advanced  far  enough 
to  understand  roughly  the  build  and  functions  of  the  eye,  it  appeared 
evident  that  the  impression  cast  through  the  lens  of  the  eye  upon  the 
sensitive  surface  was, — like  that  seen  on  the  ground-glass  focusing- 
plate  of  a  camera — inverted.  To  many  men  this  seemed  an  extraordinary 
fact.  It  yielded  for  their  minds  a  fundamental  problem :  to  show  by 
what  means  the  image  of  the  eye  was  turned  back  to  its  proper  orienta- 
tion. For  we  do  not  see  things  upside  down  at  all.  Consequently  either 
the  retinal  image  must  be  so  transmitted  to  the  brain  as  to  arrive 
there  right  side  up,  or  the  soul  itself  must  give  us  a  properly  adjusted 
view  for  the  inverted  impression  it  receives.  Echoes  of  this  kind  of 
reasoning  may  be  found  in  books  of  no  distant  date. 

One  philosophic  answer  to  this  problem  seemed  to  make  it  ridiculous. 
That  was  the  claim  that  it  was  here  a  question  not  of  absolute,  but 
only  of  relative  positions.  As  all  our  vision  is  'inverted,'  none  of  it  is. 
If  the  whole  world  expanded  suddenly  threefold  or  if  time  shrivelled 
to  twice  its  present  rate,  we  should  none  of  us  be  aware  of  the  change. 
So  it  is  a  matter  of  indifference  whether  visual  impressions  reach  the 
brain  erect  or  sloped  or  inverted,  so  long  as  they  are  all  modified  in  the 
same  way.  For  all  we  know  they  may  be  distorted  in  the  strangest 
ways  in  the  process  of  being  accommodated  to  the  zig-zag  turns  of  the 
cerebral  convolutions.  A  certain  eminent  writer  has  even  pointed  out 
that  in  our  field  of  vision  there  is  no  trace  of  the  holes  and  slits  in  the 
neural  continuity  that  must  correspond  to  blood  vessels  and  connective 
tissue  and  such  like.  He  considers  this  to  be  an  anomalous  feature  in 
any  systematic  co-ordination  of  brain  and  mind. 

The  philosophic  ridicule  of  this  problem  of  inversion  is  just  and 
proper  as  far  as  it  goes.  In  the  first  instance,  or  primitively,  as  it  were, 
it  is  a  matter  of  indifference  how  the  visual  field  is  orientated,  if  indeed 
it  can  be  said  to  have  an  orientation  to  anything  outside  itself  at  all. 
The  difficulty  of  any  such  absolute  orientation  may  be  illustrated  in  the 
terms  of  popular  metaphysics.  That  supposes  very  often  that  besides 
body  and  mind  we  have  a  soul.   And  perhaps  the  soul  has,  or  possesses. 


CH.  VIII]  THE  AUDITORY  FIELD  49 

the  mind.  In  the  opinion  of  some  it  is  the  soul  that  gives  us  what  we 
find  'in  our  minds,'  that  is  to  say,  our  experiences.  The  body  somehow 
acts  upon  the  soul  and  the  soul  responds  in  its  own  unique  and  scien- 
tifically incomprehensible  way.  If  so,  then  what  is  the  exact  place 
occupied  by  the  soul?  Is  it  in  the  brain,  or  at  the  brain?  The  question 
seems  absurd  to  some.  They  say  the  soul  has  no  place.  It  can  even 
be  acted  upon  from  two  places  at  once,  e.g.  from  the  two  eyes  or  the 
two  ears,  and  it  then  responds  by  giving  us  unitary  experience.  But, 
nevertheless,  for  all  we  know  the  soul  might  be  far  away  from  the  body, 
say  in  the  star  Sirius,  so  long  as  an  arrangement  had  been  made 
whereby  it  should  be  acted  upon  by  the  particular  human  body  on  our 
planet  that  belongs  to  it.  The  problem  of  the  absolute  orientation  of 
the  visual  field  is  just  as  insoluble  as  this  problem  of  the  soul's  distance 
from  the  body.  Vision  has  no  absolute  orientation  to  anything  that 
could  ever  be  discovered  by  us. 

But  has  it  not  an  orientation  towards  the  soul?  Suppose  you  invert 
the  printed  page  your  eyes  are  now  fixed  upon,  and  try  to  read  it.  If 
your  soul  may  not  be  disconcerted  by  the  change,  your  faculty  of  reading 
will  surely  feel  a  difficulty.  The  disturbance  is  almost  as  great  when 
the  page  is  turned  through  a  quarter  circle.  One  of  the  devices  often 
used  in  order  that  differences  of  colour  may  be  more  striking  and  clear, 
is  to  bend  down  so  that  the  head  is  inverted  and  to  view  the  object 
or  the  landscape  from  this  unusual  position.  We  may  even  arrange  in 
this  way  that  nothing  in  the  whole  field  of  vision  remains  uninverted, 
or  visible  in  its  uninverted  relation,  not  even  our  own  cheeks  and  eye- 
brows. Whatever  may  have  been  the  case  in  the  absolute  beginning, 
there  can  be  no  doubt  that  we,  brain  or  soul  or  both,  do  get  accustomed 
to  one  mode  of  presentation. 

An  American  psychologist  threw  much  light  on  this  question  by 
wearing  for  many  days  in  succession  glasses  that  inverted  the  whole 
of  his  visual  field.  "The  first  effect  was  to  make  things,  as  seen,  appear 
to  be  in  a  totally  different  place  from  that  in  which  they  were  felt. 
But  this  discord  between  visual  and  tactual  positions  tended  gradually 
to  disappear;  not  that  the  visual  scene  finally  turned  to  the  position 
it  had  before  the  inversion,  but  rather  the  tactual  feeling  of  things 
tended  to  swing  into  line  with  the  altered  sight  of  them.  The  observer 
came  more  and  more  to  refer  his  touch  impressions  to  the  place  where 
he  saw  the  object  to  be;  so  that  it  was  clearly  a  mere  matter  of  time 
when  a  complete  agreement  of  touch  and  sight  would  be  secured  under 
these  unusual  conditions.    And  when  once  the  sight  of  things  and  the 

W.  F.  M .  4 


50  OUR  POINT  OF  VIEW  [ch. 

feeling  of  them  accord  perfectly,  then  all  that  we  mean  by  upright 
vision  has  been  attained"  (63,  147). 

From  this  important  experiment,  so  trying  to  the  patience  of  the 
experimenter,  we  must  infer  that,  even  if  the  visual  field  has  no  absolute 
orientation,  it  has  at  least  a  correlation  with  the  other  sensory  fields. 
Visual  'up'  is  connected  in  our  minds  with  muscular  'up,'  and  so  on. 

But  there  must  be  still  more  than  this.  The  particles  or  minimal 
spots  of  which  any  sensory  field  may  properly  be  held  to  consist,  are 
both  absolutely  and  relatively  different  from  one  another.  This  difference 
applies  to  their  ordinal  attribute.  The  particles  of  sight  we  call  'up,' 
are  ordinally  what  they  are;  they  have  an  absolute  differentia  inherent 
in  them.  This  order  of  theirs  we  connect  by  association  with  a  muscular 
particle  (or  a  series  leading  thereto)  which  has  also  an  absolute  'order' 
of  its  own.  But  should  circumstances  suggest  it,  we  are  free  to  change 
this  connexion  by  association,  so  that  another  visual  particle,  ordinally 
very  different,  will  come  to  be  correlated  with  the  muscular  'up.'  And 
so  on.  Not  that  these  orders  are,  as  it  were,  absolute  places  in  the 
universe.  But  they  cannot  be  called  merely  relative,  because  it  is  not 
thinking  alone  that  gives  them  their  order  towards  one  another.  They 
come  to  our  thought  already  ordered;  they  are  already  such  that  if 
and  when  we  gather  them  together,  we  shall  see  that  they  actually  form 
a  system.  That  is,  their  order  is  inherent  in  each  of  them ;  it  is  absolute. 

But  this  does  not  prevent  us  from  thinking  these  orders  in  relation 
to  one  another  and  abstracting  from  their  absolute  basis  or  from  the 
associations  that  rest  upon  the  latter.  We  can  turn  a  triangle  or  a  square 
about  in  the  visual  field  so  that  it  takes  up  almost  any  sort  of  orientation 
within  or  upon  the  absolute  constituents  of  that  field.  And  so  we  learn 
to  think  a  square  and  a  triangle  independently  of  its  orientation.  But 
if  we  do  not  have  occasion  to  make  these  variations,  we  shall  learn 
to  see  a  figure  and  even  to  recognise  it  best,  or  perhaps  only,  when  it 
is  placed  a  certain  way  up.  Sometimes  we  cannot  easily  make  these 
variations.  In  other  cases  there  are  advantages  in  avoiding  them;  for 
one  and  the  same  figure — from  the  point  of  view  of  that  figure  only — 
may  become  several  figures,  if  presented  in  certain  fixed  orientations 
and  associated  mth  different  meanings  in  each  case.  Thus  h  and  y 
(as  written)  are  almost  inversions  of  one  another,  and  yet  they  are  used 
as  signs  of  different  sounds.  So  are  many  other  pairs  of  letters.  The 
advantages  are  here  all  in  favour  of  letting  the  absolutist  tendencies 
of  visual  orientation  prevail. 


vin]  TOWAKDS  THE  AUDITORY  FIELD  51 

Now  all  this  kind  of  thing  may  be  in  general  quite  familiar  in  vision. 
In  hearing,  however,  where  something  similar  occurs,  both  the  facts 
and  their  explanation  are  probably  much  less  familiar. 

The  pitch  of  a  chord  that  is  perfectly  stationary  and  is  not  at  the 
moment  apprehended  as  part  of  a  melodic  sequence,  is  most  frequently 
felt  to  be  the  pitch  of  its  lowest  component,  even  when  that  is  not  the 
strongest.  The  attention  seems  to  fall  most  easily  upon  the  lowest 
tone.  It  may  certainly  be  directed  by  melodic  means  or  by  voluntary 
effort  upon  any  other  component  of  the  whole  sound,  whether  that  be 
a  primary  tone  or  an  upper  partial  or  a  difference  tone,  or  the  like. 
But  left  to  itself  and  unguided  it  falls  back  upon  the  lowest  component, 
if  it  is  not  too  weak.  It  will  even  fall  without  instruction  upon  the 
lowest  difference-tone  that  may  be  present. 

Amongst  the  ancient  Greeks  it  appears  that  the  instrumental 
accompaniment  was  always  above  or  higher  in  pitch  than  the  melodic 
voice.  "In  the  twelfth  problem  it  is  explicitly  stated  that  the  voice 
occupies  the  lower  part  of  the  harmony.  'Why  does  the  lower  of  two 
notes  always  take  up  the  melody?^...'  There  are  extant  in  Plutarch 
two  texts  no  less  decisive  of  which  the  first  is  :  'What  is  the  cause  of 
consonance  and  why,  when  consonant  sounds  are  struck  simultaneously, 
does  the  melody  belong  to  the  lower?'  And  the  second  :  'In  the  same 
way  as,  if  two  consonant  sounds  are  taken,  it  is  the  lower  that  makes 
the  song.'  The  custom  of  putting  the  accompaniment  higher  seems 
to  have  maintained  itself  during  the  Roman  period"  (14,364)..  The 
accompaniment  might  descend  to  unison  with  the  voice,  but  not  go 
below  it.  In  early  Western  music  the  vox  principalis  was  at  first  higher 
than  the  vox  organalis,  but  after  a  time  it  took  the  lower  place  and 
remained  there  (81,  96). 

This  constant  obviousness  of  the  lowest  tone  is  an  important  factor 
in  music,  where,  as  Macfarren  said,  the  bass  "is  always  the  most 
sonorous  part  in  the  harmony"  (35,  99).  It  is  to  be  explained,  as 
already  indicated,  by  the  fact  that  the  lowest  tone  includes  all  simul- 
taneous higher  tones  within  its  volume,  and  that  the  pitch  of  the 
lowest  component  is  the  central  point  of  the  whole  tonal  mass  of  any 
moment.  Tone  is  specifically  balanced  or  graded  volume  of  sound. 
In  so  far  as  we  apprehend  sounds  tonally  at  all,  we  must  look  at 

^  Instead  of  "  melody  "  Stumpf  understands  in  the  first  place  "  pitch  "  (65,  i9).  Compare 
the  suggestions  raised  by  my  conclusions  below.  Chap,  xvni,  end.  In  connected  music 
the  more  prominent  pitch  of  an  isolated  interval  would  become  the  moro  prominent 
melody. 

4—2 


52  OUR  POINT  OF  VIEW  [ch. 

them  centrally,  as  it  were.  Thus  whatever  else  we  may  observe  in  a 
tonal  mass,  if  we  apprehend  it  as  a  whole,  we  shall  inevitably  look  at 
it  centrally  and  so  most  readily  come  upon  the  pitch  of  its  largest 
volume,  i.e.  of  the  lowest  component. 

This  prevailing  attitude  shows  itself  in  a  number  of  other  ways. 
We  agree  in  reckoning  the  interval  between  any  two  pitches  upwards, 
unless  some  special  indication  to  the  contrary  is  given.  Thus  C-E  is  to 
be  taken  as  a  major  third,  not  as  a  minor  sixth.  Even  the  Greeks, 
who  in  practice  found  it  more  fitting  to  pass  from  high  to  low  than 
conversely,  reckoned  all  intervals  theoretically  from  below  upwards 
(16,  89;  14,  173  ff.).  Rising  of  pitch  gives  us  the  impression  of  departure, 
lowering  of  pitch  that  of  approach.  We  incline  to  take  a  scale  from 
below  upwards  and  back  again,  rather  than  downwards  and  up  again 
to  the  starting-point.  In  a  major  chord  we  consider  the  major  third 
as  the  first  of  the  two  intervals,  the  minor  third  as  the  second.  The 
tonic  of  a  major  chord,  whether  its  component  tones  are  given  succes- 
sively or  simultaneously,  is  held  to  be  the  lowest  of  the  three.  The 
attempt  has  been  made  to  look  upon  the  highest  tone  of  the  minor 
triad  as  its  root  or  tonic.  But  the  very  strangeness  of  the  claim,  apart 
from  the  validity  of  the  special  arguments  advanced  in  its  favour, 
shows  that  it  does  not  correspond  to  our  actual  attitude  towards  the 
chord.  Moreover,  when  an  interval  is  mistaken  for  a  single  tone,  as  in 
the  experiments  on  fusion,  the  pitch  ascribed  to  it  is  in  the  majority 
of  cases  that  of  the  lower  tone. 

Another  strong  evidence  of  this  central  attitude  to  tonal  groups  is 
found  in  certain  striking  differences  between  ascending  and  descending 
intervals  involving  the  same  tones.  "The  most  of  those  who  can 
recognise  intervals  at  all  have  learnt  in  the  first  place  to  judge  them 
in  the  ascending  form.  The  estimation  of  descending  intervals  is  much 
harder;  in  fact  it  is  primarily  quite  a  different  task....  The  difficulty 
of  judging  descending  intervals  appears  not  only  amongst  less  practised 
observers,  but  also  amongst... persons  who  had  all  enjoyed  a  good 
musical  education.  In  judging  descending  intervals  indirect  criteria 
were  often  used  by  them.  The  time  spent  in  recognising  these  intervals 
was  also  larger  than  that  required  for  ascending  intervals"  (37,  192). 

This  may  seem  at  first  to  be  a  very  extraordinary  fact,  hardly 
creditable  by  those  who  recognise  all  intervals  at  once  without  hesitation 
or  by  those  who  find  difficulty  in  naming  any  by  ear  alone.  It  is  certainly 
incompatible  with  any  purely  relativistic  interpretation.  For  if  the 
relation  a  to  6  is  recognisable,  the  relation  of  6  to  a  should  be  so  also 


vm]  TOWARDS  THE  AUDITORY  FIELD  53 

as  a  matter  of  course,  since  it  is  the  same  relation.  But  if  a  point  of 
view  has  been  adopted  and  if  a  and  h  are  not  of  a  purely  qualitative 
nature,  we  can  readily  understand  that  the  appearance  of  a-h  from 
the  standpoint  of  6  may  be  very  different  from  its  appearance  from 
the  position  a.  The  face  of  a  friend  seems  very  strange  when  it  is  seen 
inverted.  Even  the  letters  of  our  alphabet  or  simple  ornamental  figures 
become  unfamiliar  then. 

This  peculiar  difference  produced  by  the  direction  of  interval  is 
seconded  by  a  similar  distinction  between  simultaneous  and  successive 
intervals.  When  musically  untrained  persons  have  been  taught  to 
attach  the  correct  names  to  simultaneous  intervals,  it  is  found  that 
they  are  quite  incapable  of  naming  the  corresponding  successive  forms, 
in  spite  of  great  practice  at  the  former  task.  They  cannot  mentally 
convert  the  succession  into  simultaneity  (37,  192). 

We  may  therefore  look  upon  it  as  well  founded  that  we  do  adopt 
a  strangely  prevaihng  attitude  towards  the  tonal  series.  Our  point 
of  view  is  for  any  moment  that  of  the  centre  of  the  whole  tonal  mass. 
This  centre  need  not,  of  course,  merely  be  the  centre  of  a  single  tone 
or  of  the  momentary  mass  of  sound  that  is  at  the  ear.  It  may  be  the 
centre  of  a  tonal  complex  begun  a  moment  ago  and  lasting  on  for  a 
span  of  time  till  it  is  completed.  How  long  this  span  may  be,  will 
depend  greatly  upon  our  musical  practice  and  upon  the  musical  coherence 
and  stabihty  of  the  complex  that  is  presented.  These  complexes  vary 
in  length  and  complexity  very  much.  It  is  also  possible  that  in  viewing 
the  various  parts  of  this  complex  as  they  flow  past  us,  we  do  not  need 
to  maintain  the  central  position  that  is  valid  for  the  whole  complex 
in  any  rigid  way,  so  long  as  our  disposition  towards  it  remains  ready 
and  active.  We  may  then  wander  about  with  the  centres  of  each 
momentary  sound  mass,  always  having  it  in  our  power  to  see  the 
relation  of  that  to  the  general  centre  of  the  whole  and  to  return  to  the 
latter  if  required.  All  this  would,  of  course,  not  be  an  inevitable  and 
unshakable  consequence  of  the  primary  centrality  of  tone,  but  would 
gradually  develop  out  of  it  by  the  practice  and  mental  skill  of  the 
listener  and  by  the  support  given  to  him  by  the  devices  of  musical  art. 
Thus  we  see  how  the  primary  point  of  view  towards  tone  might  develop 
towards  the  special  point  of  view  we  know  in  music  as  tonality,  the 
feeling  for  a  tonic,  a  point  of  reference  for  the  tones  and  chords  of  a 
musical  unit. 

Nor  does  the  fact  that  the  central  point  of  view  towards  tone  is 
the  natural  and  prevalent  attitude  prevent  us  from  acquiring  another 


54  THE  AUDITORY  FIELD  [ch.  vm 

point  of  view  by  special  practice  or  preference.  Those  who  sing  a 
certain  part  in  songs  or  hymns  regularly  and  whose  musical  practice 
is  predominantly  of  this  kind,  will  doubtless  find  it  easier  to  follow 
their  usual  habit.  The  musical  analysis  of  many  persons  is  confined 
to  attention  to  soprano  melodies.  Even  if  their  analysis  goes  beyond 
this,  their  greatest  practice  and  interest  may  yet  tend  oftenest  to  the 
highest  voice,  so  that  if  a  single  chord  be  given  they  will  select 
from  it  as  its  pitch  its  highest  (primary)  component.  Modern  music 
teaches  everyone  to  pay  special  attention  to  soprano  melody.  For  it 
commonly  endeavours  to  put  the  maximum  of  interest  into  one  such 
melody  and  subordinates  the  melodic  interest  of  other  voices  to  their 
harmonic  beauty,  D.  F.  Tovey  expresses  this  when  he  defines  melody 
as  "the  surface  of  music"  (75).  Our  present  interest  is  not  to  investigate 
or  to  depreciate  the  importance  of  any  such  special  points  of  view 
towards  music,  but  only  to  show  how  various  facts  indicate  that  the 
natural,  original  or  fundamental  point  of  view  is  a  central  one,  or 
extends  from  a  variable  point  or  centre  upwards  in  the  tonal  range. 

At  the  same  time  these  facts  support  strongly  the  theory  of  the 
volumic  proportional  nature  of  interval  and  bring  the  study  of  tone 
and  music  into  most  intimate  agreement  with  facts  that  better  natural 
endowment,  the  greater  scope  of  physical  variation  and  greater  practice 
have  made  more  or  less  familiar  to  us  all  in  vision. 


CHAPTER  IX 

THE  RELATIVE  IMPORTANCE  OF  SYNTHESIS  AND  OF  ANALYSIS 

We  have  now  gone  so  far  as  to  be  able  to  look  back  upon  the  field 
of  tone  and  to  survey  it  somewhat  as  a  whole. 

The  opinion  has  been  often  expressed  that  science  can  never  give 
a  proper  account  of  any  art,  because  the  aim  of  science  is  analytic; 
it  strives  to  dissect  and  to  divide,  tracing  each  part  to  its  separate 
root  and  origin.  It  must  necessarily  lose  the  life  and  spirit  of  the  whole. 
No  doubt  this  is  true  so  long  as  a  science  is  busy  over  the  preliminary 
efforts  of  analysis  and  has  not  yet  reached  the  stage  of  tracing  the 
synthesis  that  binds  the  many  parts  together.  But  analysis  is  not  the 
final  condition  in  which  scientific  results  are  to  be  left. 

The  study  of  the  body  involves  a  long  course  of  special  study  of 
each  distinguishable  part  and  of  its  own  particular  functions.  It  is 
only  thus  that  we  can  learn  what  primary  functions  or  processes  are 
at  work  in  the  living  body.  And  it  is  only  from  the  basis  of  this  knowledge 
that  we  can  venture  to  explain  the  united  work  or  the  integrative 
action  of  the  living  body. 

The  study  of  the  mind  at  first  calls  for  a  minute  examination  of 
every  distinguishable  experience,  its  fundamental  variability  and  its 
primary  relations  to  other  experiences.  But  the  science  of  the  mind 
is  not  to  be  taken  as  a  mere  catalogue  of  pieces  and  processes  without 
connexion  with  one  another.  That  would  be  to  mistake  its  achievements 
at  a  certain  early  period  of  its  development  for  the  results  it  may  in 
the  course  of  time  properly  expect  to  attain.  One  of  its  duties  is  to 
aspire  to  show  how  the  mind  of  the  average  man  appears  to  him  as 
it  does  and  why. 

In  the  same  way  the  science  of  music  has  first  to  dig  down  to  its 
foundations  and  show  their  form  and  connexions.  Only  then  can  it 
build  upwards  from  these  and  aspire  to  give  a  full  and  true  account 
of  music  as  it  appears  to  the  musical  mind,  i.e.  to  the  mind  that  is  not 
crowded  with  scientific  knowledge  concerning  music  and  actually 
thinking  of  it,  but  to  the  mind — even  though  it  be  the  same  mind  or 
person — that  is  for  the  moment  hearing  and  enjoying  music  in  the 
ordinary  way. 


56  THE  RELATIVE  IMPORTANCE  OF  SYNTHESIS        [ch. 

The  science  of  music  has  for  centuries  paid  the  greatest  attention 
to  harmonics  or  upper  partial  tones.  It  has  tried  to  explain  many 
things  by  them.  The  insufficiency  of  the  results  has  turned  the  hopes 
of  theorists  in  later  years  towards  the  lower  tones  that  appear  in  chords, 
namely  to  the  difEerence-tones.  But  of  all  these  things  the  ordinary 
musical  mind  is  quite  regardless  in  hearing  and  enjoying  music.  It  is 
only  with  difficulty  and  effort  that  it  can  be  brought  to  recognise  their 
existence  even  when  the  attention  is  not  aesthetically  engaged.  And 
when  it  is  again  so  occupied,  harmonics  and  difEerence-tones  disappear 
from  view  entirely.  That  fact  alone  suggests  the  view  that  harmonics 
and  difference-tones  have  not  the  central  importance  for  musical  theory 
that  has  often  been  claimed  for  them.  And  it  confirms  a  theory  that 
can  find  other  foundations  of  greater  validity. 

This  does  not,  however,  mean,  as  some  have  seemed  to  think,  that 
the  scientific  attention  creates  harmonics,  or  that  harmonics  come  and 
go  according  to  the  inclination  of  observation.  I  say  'seemed  to  think,' 
for  no  one  can  venture  to  maintain  such  a  view  outright.  We  may 
want  for  our  satisfaction  to  think  that  "the  self  sets  itself"  first  and 
then  all  the  rest  of  the  world,  including  harmonics,  according  to  its 
inclinations  of  self-realisation.  If  it  is  possible  to  leave  this  marvellous 
power  to  a  Universal  Self,  we  may  well  do  so.  But  for  our  own  self  we 
must  refuse  to  believe  that  it  is  able  by  mere  change  of  attention  to 
set  anything  into  being  at  all.  If  harmonics  are  there  when  we  attend 
to  them,  then  they  are  also  there  when  we  do  not  attend  to  them. 
What  we  have  to  explain  is  why,  when  we  attend  to  them,  they  appear 
in  a  different  way  than  they  otherwise  do. 

And  that  is  not  a  difficult  task.  When  we  attend  to  a  harmonic, 
we  concentrate  our  inward  gaze  upon  it  alone  to  the  exclusion  of  any 
setting  or  circumstances  it  may  stand  in.  So  we  notice  its  own  particular 
pitch  and  we  can  form  a  fairly  sufficient  estimate  of  its  own  particular 
volume.  We  may  arrange  for  the  independent  production  of  a  tone 
very  like  it  and,  by  noticing  the  beating  of  the  latter  with  the  harmonic, 
estimate  its  pitch  precisely.  But  we  do  not  create  it  by  our  attention. 
For  we  have  no  knowledge  from  our  will  alone  what  its  properties  will 
be  and  real  tones  will  not  beat  with  fancied  ones. 

When  we  cease  to  attend  specially  to  harmonics  they  are  in  them- 
selves quite  unaltered  thereby.  But  if  we  then  attend  to  the  tone  that 
contains  them,  we  hear  them  in  their  full  setting;  we  hear  them  as 
a  part  of  the  tone  or  chord  we  are  attending  to.  And  then,  as  everyone 
knows,  they  appear  to  us  as  the  particular  blend  (or  timbre)  of  the  tone. 


IX]  AND  OF  ANALYSIS  57 

They  do  so  because,  being  higher  than  the  primary  tone  which  gives 
the  whole  tone  its  musical  pitch,  they  all  fall  within  its  volume.  And 
in  good  musical  tones,  the  upper  partials  are  of  a  restricted  intensity, 
wherefore  they  do  not  stand  out  prominently  in  the  volume  of  the 
whole  tone  so  as  to  call  the  attention  specially  to  themselves.  They 
leave  the  balance  and  symmetry  of  the  fundamental  still  obvious. 
These  qualities  are  no  longer  so  perfect,  of  course,  as  are  those  of  the 
pure  tone.  But  they  are  far  from  being  so  vague  and  deteriorated  that 
the  sound  could  be  mistaken  for  noise,  in  which  balance  and  symmetry 
have  been  lost  or  at  least  made  very  hard  to  find. 

The  harmonics  of  a  good  musical  tone  only  make  a  slight  change 
of  surface,  as  it  were,  in  the  whole  tone.  It  no  longer  remains  perfectly 
smooth  like  the  pure  tone;  its  volume  acquires  a  character  whose 
nature  depends  upon  the  harmonics  present.  A  set  of  very  high  harmonics 
will  give  the  tone  a  touch  of  highness  or  brightness.  The  lower  harmonics 
will  give  more  variety  to  the  central  body  of  the  tone;  it  will  not  be 
empty  and  poor,  like  the  pure  tone,  but  full  of  interest  and  rich.  If 
only  the  uneven  numbered  partials  occur,  the  tone  will  take  on  another 
character,  one  that  appears  in  the  sounds  produced  by  the  nasal  voice 
and  by  hollow  cavities  of  various  kinds,  so  that  we  associate  the  idea 
of  hollo wness  with  it,  and  call  it  a  hollow  sound.    And  so  on. 

The  interests  of  music  are  not  commonly  served  by  sounds  in  which 
partials  attract  the  attention  to  themselves  or  are  separately  distin- 
guishable with  ease.  Those  tones  are  the  most  valuable  in  which  the 
minimal  reduction  of  smoothness  is  compensated  by  a  maximal  richness 
and  interest  of  pitch-blend.  Tone  must  be  rich  and  strong  without 
being  rough,  and  smooth  without  being  dull  or  poor.  It  should  be  at 
once  as  full  and  as  rich  as  may  be.  In  analytical  terms,  the  fundamental 
must  be  present  in  good  strength  to  give  the  tone  a  fullness  of  the  volume 
it  '  aspires '  to  or  is  meant  to  be ;  and  a  typical  series  of  partials  should 
'colour'  it  or  give  it  a  characteristic  surface  without  being  so  strong 
as  either  singly  or  collectively  to  outweigh  the  fundamental  or  to  stand 
forth  in  it  so  much  that  they  take  upon  themselves  the  rank  of  primary 
sounds — tones  actually  played  separately  by  the  performer  and  written 
by  the  composer  or  intended  to  be  heard  separately. 

It  is  unnecessary  to  recall  that  great  variety  of  beautiful  tone- 
surface  is  of  the  highest  importance  in  music.  These  blends  give  a  new 
interest  to  repetition  and  a  new  line  of  variation  by  which  the  hearer's 
mind  may  be  led  to  give  ear  to  the  secret  of  the  soul's  life  that  the  artist 
strives  to  convey. 


58  THE  RELATIVE  IMPORTANCE  OF  SYNTHESIS        [ch. 

The  musical  attitude  towards  harmonics,  then,  is  the  synthetic 
attitude.  They  create  beauty  when  their  synthesis  is  easy  or  inevitable, 
i.e.  when  their  strength  is  so  graded  and  unobtrusive  that  they  appear 
to  the  attention  only  as  a  minor  modification  of  the  tonal  volumes 
that  compel  the  attention.  But  when  the  attention  is  used  analytically — 
as  when  we  pass  from  stone  to  stone  of  an  architectural  surface  or  from 
stroke  to  stroke  of  the  brush  in  a  picture — harmonics  can  be  inspected 
singly.  The  rest  of  the  tonal  mass  tends  to  disintegrate.  The  attention 
is  then  concentrated  on  the  part  and  is  scattered  in  the  rest;  whereas 
in  a  synthetic  unity  such  as  an  artistic  object  the  centre  of  attention 
is  so  placed  that  it  radiates  easily  to  the  parts  and  binds  them  together 
in  itself,  while  they  point  towards  it  and  so  make  it  easy  to  find  rapidly. 

Let  us  now  consider  difference- tones  in  the  same  relations.  These 
lower  partials,  as  it  were,  do  not  accompany  single  tones,  so  that  they 
cannot  play  the  same  part  in  giving  blend  or  surface  to  a  tone  as  upper 
partials  do.  They  appear  only  when  at  least  two  primary  tones  are 
sounded.  Of  course,  they  must  also  be  produced  by  the  interaction  of 
upper  partials  as  primaries.  But  the  artistic  subjection  of  these  to  the 
fundamental  partial  does  not  allow  of  their  usually  appearing  in  a  single 
blended  tone  in  any  noticeable  degree.  Difference-tones  are  in  any 
case  weaker  than  their  primaries,  so  that  if  they  originate  from  the 
partials  of  a  blend,  they  will  be  weaker  than  these  and  will  therefore 
have  less  chance  of  being  separately  noticed  in  a  blended  tone  than 
have  its  partials.  Besides,  the  partials  of  the  harmonic  series  could 
never  produce  either  a  difference-tone  that  was  lower  than  the  funda- 
mental of  that  series  or  that  did  not  coincide  with  some  member  of 
the  theoretical  harmonic  series.  So  any  new  component  of  a  sound 
that  was  produced  as  the  difference-tones  of  its  partials  would  only 
appear  to  be  another  partial  of  that  sound. 

The  difference-tones  that  are  due  to  primary  tones  are  of  quite 
considerable  strength.  This  is  true  at  least  for  the  first  difference-tone 
(higher  rate  of  vibration  minus  the  lower,  oi  k  —  I)  and  for  the  second 
difference-tone  {21  —  h).  The  other  difference-tones  are  much  weaker 
and  very  difficult  to  hear,  so  that  their  inaudibility  to  the  ordinary 
ear  under  usual  circumstances  hardly  forms  a  problem  for  any  possible 
theory  of  sound.  In  spite  of  their  loudness  these  first  two  difference- 
tones  are  much  less  easily  noticed  than  partials — as  the  much  later 
discovery  of  them  shows.  Reasons  for  this  obscurity  are  not  far  to  seek. 
The  strongest  is  the  nature  of  their  origin.    They  appear  only  when 


IX]  AND  OF  ANALYSIS  59 

two  sounds  are  played  together.  In  order  to  detect  them  one  has  to 
consider  carefully  what  exactly  each  of  these  two  sounds  separately 
contains  and  then  to  subtract  the  sum  from  what  is  heard  when  both 
are  played  together.  Most  persons  do  not  trouble  to  do  this.  They 
expect  the  tonal  mass  of  the  two  sounds  to  contain  more  than  that 
of  either,  as  it  obviously  does.  But  the  peculiar  overlapping  and  blending 
of  simultaneous  tones  prevents  them  in  most  cases  from  discerning 
precisely  the  exact  contribution  of  each  of  the  primary  components. 
Thus  the  new  whole  passes  as  the  peculiar  mixture  of  the  primaries. 
This  tendency  is  encouraged  by  the  fact  that  in  the  octave  where  the 
fusion  of  the  primaries  gives  the  simplest  product,  there  is  only  one 
difference-tone  which  is  identical  with  the  lower  primary.  So  then 
where  one  might  most  readily  have  detected  an  addition,  there  is  nothing 
new  to  find.  And  in  the  fifth,  the  only  difference-tone  is  exactly  an 
octave  below  the  lower  primary  and  so  fuses  with  it  as  much  as  any 
tone  could,  thus  making  detection  again  difficult.  A  third  reason  lies 
in  the  fact  that  difference-tones  are  not  under  ordinary  circumstances 
found  to  exist  outside  the  ear,  so  that  they  could  not  be  discovered, 
as  were  so  many  of  the  chief  facts  of  acoustics,  from  a  study  of  the 
movements  and  resonance  of  the  sonorous  body.  They  had  to  be 
found  purely  by  the  inspection  of  sound  itself.  And  the  attention  was 
naturally  directed  in  that  to  the  primary  sounds  that  were  intended 
and  played  and  upon  which  musical  structure  primarily  rests. 

Nevertheless  difference-tones  did  not,  of  course,  first  come  into  being 
at  their  discovery.  They  were  there  all  along,  moulding  the  character 
of  chords  as  a  whole,  giving  them  especially  a  touch  of  a  largeness  of 
volume  that  their  primaries  did  not  contain.  The  listener,  as  is  often  said, 
is  unconsciously  affected  by  them.  That  does  not  mean,  to  be  sure, 
that  his  body  or  brain  is  affected  by  them,  but  not  his  mind.  It  means 
only  that  while  the  lowness  is  there  in  his  sensations  in  a  particular 
form,  and  he  hears  it  as  a  lowness  appertaining  to  the  whole  sound, 
yet  he  does  not  separate  it  out  in  its  discrete  form  in  the  whole  and 
know  it  as  such.  It  would,  therefore,  be  better  to  say  that  the  listener 
is  unwittingly  affected  by  the  difference-tones.  For  if  sensation  is  a 
form  of  consciousness,  and  the  difference-tones  are  in  sensation,  the 
hearer  is  of  course  conscious  of  them.  If  sensations  are  considered  to 
be  objects  presented  to  the  mind,  which  is  only  conscious  of  them 
when  it  knows  them  individually,  then  the  hearer  is  unconscious  of 
difference-tones  even  when  he  attends  to  the  sound  complex  in  which 
they  appear  before  him,  until  he  has  separated  them  out  from  the  group 


60  THE  RELATIVE  IMPORTANCE  OF  SYNTHESIS        [ch. 

of  sensory  objects  presented  to  him  and  has  cognised  them  individu- 
ally. 

Having  thus  surveyed  the  outlying  components  of  a  sound  mass, 
we  may  now  deal  with  the  relative  importance  of  synthesis  and  analysis 
amongst  the  primaries. 

It  is  to  be  noted  first,  however,  that  the  objection  brought  against 
harmonics  and  difference-tones  as  the  foundations  and  regulators  of 
musical  structure  does  not  hold  for  our  interpretations  of  the  primaries. 
That  objection  is  that  the  composer  and  the  hearer  commonly  know 
nothing  about  harmonics  and  difference-tones  and  care  still  less,  least 
of  all  when  they  are  actually  in  the  aesthetic  attitude.  They  neither 
know  of  these  things  nor  do  they  attend  to  them.  But,  while  they  have 
certainly  not  Jcnown  about  the  volumes  and  coincidences  and  proportions 
of  tones  either,  they  have  indeed  always  attended  to  them.  For  tone 
and  interval  are  not  derived  from  volume  and  proportion  as  from  things 
that  lie  in  a  land  beyond  their  own  :  on  the  contrary,  they  are  volume 
and  proportion.  Whoever  attends  to  tone  and  to  interval,  attends  to 
volume  in  its  balance  and  symmetry  and  to  proportion  of  volumes. 
What  our  theory  has  done  is  neither  to  trace  the  heredity  of  tone  and 
of  interval  and  of  fusion,  nor  to  say  what  remote  stars  have  influenced 
their  horoscope;  but  it  has  dissected  the  very  body  of  these  things, 
as  it  were,  showing  what  they  consist  of  and  how  they  are  related  to 
one  another  and  to  other  similar  things.  Thus  the  change  for  the 
artist  and  hearer  is  merely  from  the  practical  and  aesthetic  attitude 
to  the  cognitive  attitude — towards  one  and  the  same  material.  To 
skill  and  feeling  is  added  knowledge.  From  using  merely  nominative 
terms  for  the  objects  of  sense  we  pass  to  systematic  terms,  which  not 
merely  point  them  out  on  the  basis  of  mental  association,  but  which 
indicate  their  place  amongst  other  objects  and  their  relations  to  them 
on  the  basis  of  systematic  knowledge.  No  objection  can  be  brought 
against  this  knowledge  from  the  artistic  or  practical  point  of  view,  for 
it  builds  upon  the  same  ground  as  they  do.  It  only  adds  the  fullness  of 
knowledge  to  the  sufficiency  of  sense.  Then  sense  not  only  is  present  with 
the  mind  and  affects  it  to  feeling  and  emotion,  but  it  is  known  as  well. 

The  primary  tones  of  a  chord  blend  with  one  another  or  with  their 
fundamental  in  the  same  general  way  as  do  harmonics.  They  are  much 
more  easily  recognised  in  the  whole  partly  because  they  are  louder, 
partly  because  they  are  known  and  intended  to  be  played.  They  are 
part  of  the  player's  conscious  intention,  just  as  the  blend  or  surface 


ix]  AND  OF  ANALYSIS  61 

of  tone  in  its  synthetic  form  is.  We  have  already  seen  how  tones  an 
octave  apart  may  fuse  so  well  together  as  to  be  mistaken  for  one.  This 
high  fusion  of  loud  tones  approximates  to  that  generally  valid  for  the 
weak  tones  of  partials. 

But  the  practised  musician  is  able  to  pick  out  the  primary  tones 
of  a  chord  with  considerable  ease.  The  most  gifted  ear  can  pick  them 
out  at  once  unfailingly  in  any  part  of  the  musical  range  and  on  any 
instrument.  This  analysis  cannot,  of  course,  annul  the  underlying 
synthesis  of  tones  that  is  due  to  their  volumic  overlapping.  But  the 
gifted  ear  can  at  once  seize  upon  the  pitch  predominances  that  the 
volumes  contain,  so  as  to  cognise  the  component  parts  of  the  whole. 
A  clear  analytic  view  is  obtained  without  any  of  the  synthetic  effects 
of  overlapping  or  fusion  being  lost.  Analysis,  at  its  best — in  dealing 
with  primaries — does  not  require  the  finest  ear  to  pass  successively 
from  one  pitch-point  to  another  in  order  to  cognise  them  all.  They 
are  all  grasped  at  once,  as  any  of  us  grasps  the  whole  of  a  simpler  visual 
pattern  or  of  a  word  in  one  gaze  of  fixation. 

Nor  is  there  here  any  general  confusion  between  primaries  and 
harmonics.  For  the  latter  are  heard  on  all  famihar  instruments  as 
synthetic  blends,  not  as  separate  tones.  If  the  grouping  of  tones  makes 
one  or  other  harmonic  very  loud,  this  will  tend  to  be  heard  as  a  primary 
tone.  But  the  prevailing  attitude  will  be  to  distinguish  only  the  loudest 
components  as  primaries  and  to  hear  harmonics  in  their  usual  blend. 
This  attitude  is  greatly  supported  by  the  expectations  made  habitual 
by  the  general  course  of  musical  spelling  and  grammar,  into  which 
harmonics  will  not  often  fit  coherently. 

The  less  gifted  ear  does  not  distinguish  the  primary  tones  so  readily. 
It  may  well  learn  to  recognise  each  interval  and  chord  as  a  whole,  as  a 
characteristic  thing,  as  one  learns  to  recognise  words  as  a  whole  without 
reading,  or  thinking  of,  each  letter  separately.  And  it  may  also  then 
readily  learn  to  spell  out  the  tones  in  the  easier  or  more  frequent 
groupings,  so  as  to  be  able  at  least  to  name  the  relative  pitches  of  each. 
But  the  first  prevailing  tendency  is  synthetic,  even  over  and  above  the 
inevitable  synthesis  of  fusion;  analysis  of  (stationary)  chords  is,  then, 
the  result  of  effort  and  special  attention. 

But  in  melody  the  attention  is  almost  relieved  of  any  effort  of  analysis. 
The  analysis  takes  place  as  a  matter  of  course ;  or  rather,  the  sequence 
of  tones  that  we  call  melody  is  purposely  so  formed  that  the  attention 
will  follow  it  easily. 


62  THE  RELATIVE  IMPORTANCE  OF  SYNTHESIS       [ch. 

In  primitive  music  there  is  commonly  only  one  singing  voice,  which 
displays  some  distinct  melodic  form  whereby  its  movements  acquire 
unity  and  interest.  In  polyphonic  music  several  voices  proceed  simul- 
taneously, each  one  being  melodically  controlled  in  this  way.  In 
harmonic  music  the  fullest  melodic  treatment  is  given  commonly  only 
to  one  voice,  sometimes  to  two  concurrently.  The  rest  of  the  tonal 
mass  of  each  moment  is  handled  synthetically,  so  that  the  listener 
apprehends  it  rather  as  a  whole.  The  sequence  of  tonal  masses  is 
regulated  partly  by  the  requirements  of  melodic  form  and  partly  by 
the  relations  connecting  harmonic  chords.  Polyphonic  music  is,  of 
course,  also  a  sequence  of  (harmonic)  chords;  but  the  melodic  treatment 
of  all  the  voices  leads  to  a  predominance  of  the  melodic  connexions 
of  the  homologous  voices  of  successive  chords  over  the  harmonic  or 
fusional  connexions  within  each  chord. 

The  degree  to  which  the  harmonic  and  melodic  aspects  of  music 
prevail  over  one  another  is  thus  very  variable.  At  the  one  extreme 
we  find  each  voice  so  perfectly  finished  melodically  that  both  the 
artist  and  the  auditor  fail  to  apprehend  the  harmonic  values  of  successive 
chords  in  any  special  way,  although  they  in  no  wise  fail  to  hear  how 
far  the  different  voices  fit  agreeably  into  one  another's  movements. 
The  basis  of  this  agreeable  conjunction  is,  of  course,  the  fusional 
relations  of  the  tones  of  each  chord.  These  are  necessarily  indestructible 
and  irremovable  by  any  treatment  of  the  attention  or  by  any  abstraction. 
But  nevertheless  a  special  attitude  of  abstraction  does  lead  the  ear 
to  make  as  little  as  possible  of  them  for  the  production  of  the  larger 
syntheses  of  the  art.  At  the  other  extreme  we  find  the  melodic  interest 
completely  subordinated  to  the  harmonic.  Each  chord  is  a  fusional 
mass  enjoyed  for  its  special  'colour'  or  mass  effect.  The  sequence  of 
chords  is  not  decided  on  the  basis  of  melodic  form  in  any  specific  sense. 
Melodic  sequence,  in  general,  is,  of  course,  just  as  insuppressible  and 
irremovable  as  is  harmony.  The  various  chords  that  follow  one  another 
must  do  so  in  such  a  way  as  to  satisfy  the  minimal  demands  of  melodic 
movement  generally.  They  must  move  by  as  small  steps  as  possible 
and  must  not  cross  one  another,  and  so  on.  This  minimum  is  enough 
to  guide  the  ear  easily  from  one  chord  to  the  next,  but  it  is  not  enough 
to  create  any  sort  of  melodic  form.  In  fact  the  melodies  that  result  may 
be  perfectly  irregular.  They  only  pro\'ide  enough  obvious  movement  to 
guide  the  ear.  Thus  the  mind  is  left  free  to  devote  itself  to  the  harmonic 
interests  of  the  music,  and  the  artist  or  improviser  may  pay  his  greatest 
attention  to  building  effects  upon  a  synthesis  of  harmonic  sequences. 


IX]  AND  OF  ANALYSIS  63 

The  matter  may  be  stated  in  a  somewhat  more  figurative  manner. 
Each  mass  of  sounds  that  constitutes  music  in  several  parts  or  voices 
has  two  aspects  :  the  one  is  its  volumic  aspect,  the  fusion  characteristic 
of  it  as  a  whole,  or  in  any  of  its  parts,  i.e.  between  any  two  of  its  voices; 
the  other  is  its  pitch  aspect  or  its  ordinal  predominances,  the  points  of 
sound  that  stand  forth  intensely  in  it.  The  art  that  builds  up  chords 
into  complex  music  may,  as  it  were,  make  either  of  these  two  aspects 
the  surface  of  the  product  that  is  to  be  exposed  to  the  hearer,  while  the 
other  is  made  the  mere  surface  of  suture,  cementing  one  brick  of  the 
building  to  another. 

If  the  ground  of  artistic  synthesis  is  pitch,  great  care  must  be  taken 
in  the  selection  of  each  brick  that  its  pitch-points  fit  into  those  of  the 
next,  so  that  the  sequence  will  give  a  perfect  complex  of  melodic  figures, 
easily  surveyed  by  the  listener.  The  subsidiary  interests  of  the  art 
require  the  sequent  chords  to  be  so  harmoniously  consistent  that  they 
will  not  severally  fall  to  pieces  or  confuse  the  movements  of  the  different 
voices  and  will  yet  knit  together  so  as  to  make  a  stable  whole.  But  the 
Mstener  is  not  concerned  with  them  beyond  this. 

If  the  ground  of  synthesis  is  harmony,  that  aspect  of  each  chord 
will  be  turned  outwards.  Sequences  will  be  selected  specially  for  the 
manner  in  which  they  link  together  to  form  large  harmonic  surfaces 
or  masses,  as  it  were.  The  melodic  aspect  is  required  only  in  so  far  as 
it  helps  to  bind  the  chords  to  one  another  on  the  unexposed  surface 
and  so  to  perfect  the  underlying  stability  of  the  structure. 

Or,  if  you  like,  in  the  one  style  of  music  harmony  is  put  in  the  focus 
of  the  listener's  conscious  mind,  while  melody  remains  in  the  background 
of  it;  in  the  other  style  conversely.  Or  again,  in  the  one  harmony  is 
merely  sensed  and  felt,  while  melody  is  built  up  into  complex  figures, 
inspected,  and  watched  in  all  its  changes,  and  consciously  enjoyed; 
in  the  other  the  melody  is  merely  sensed  as  an  atmosphere,  while  the 
specific  artistic  structure  is  harmonic. 

All  this  comes  to  the  same  thing  as  John  Hullah's  oft  repeated 
dictum  about  the  horizontal  (melodic)  and  the  perpendicular  (harmonic) 
views  of  musical  structure^.    The  figure  of  speech  is  here  derived  from 

*  V.  25,  106:  "I  use  the  word  harmony  aa  representing  the  successive  results  of  an 
accumulation  of  parts.  For  of  a  chord,  as  an  isolated  fact,  the  old  masters  took  little  account. 
They  were  not  harmonists  at  all,  in  our  sense  of  the  word,  but  contrapuntists;  laying  melody 
upon  melody,  according  to  certain  laws,  but  uncognisant  of,  or  indifferent  to,  the  effects 
of  their  combinations  as  they  successively  came  upon  the  ear.  Their  constructions  were 
horizontal,  not  perpendicular.  They  built  in  layers,  but  their  music  differs  from  most  of 
ours  as  a  brick  wall  does  from  a  colonnade,"  etc. 


64  SYNTHESIS  AND  ANALYSIS  [ch.k 

the  structure  of  the  printed  music  in  which  the  component  tones  of  a 
chord  are  written  below  or  above  one  another,  while  melodies  run  from 
left  to  right  of  the  page  through  the  tonal  points  of  each  chord.  The 
figure  is,  of  course,  not  strictly  applicable  to  what  is  heard.  For  the 
horizontal  aspect  is  not  spatial,  as  the  adjective  suggests,  but  temporal; 
the  field  of  hearing — if  the  pitch  series  (or  the  length  of  tonal  volume) 
be  called  its  perpendicular  dimension — has  no  horizontal  aspect  at 
all  as  far  as  music  is  concerned.  But  if  allowance  is  made  for  this 
discrepancy,  the  figure  is  apposite — more  so  indeed  than  its  originator 
could  have  known.  For  the  harmonic  dimension  is  really  akin  to  a 
spatial  dimension;  it  is  ordinal,  and  space  is  probably  an  ordinal 
derivative. 

We  have  thus  characterised  in  general  the  relative  importance  of 
fusion  and  of  analysis  in  music,  and  we  have  given  these  two  aspects 
of,  or  attitudes  towards,  tonal  masses  a  basis  in  the  nature  of  these 
masses  themselves  as  sounds.  In  other  words  we  have  shown  upon 
what  features  of  tones  fusion  rests  and  what  points  of  tonal  volume 
offer  themselves  for  special  analytic  attention.  In  freely  planned 
experiments  these  attitudes  may  be  prescribed,  or  imposed  upon  oneself 
voluntarily  and  followed  at  leisure.  As  in  other  regions,  so  here  it  is 
found  that  some  circumstances  make  synthetic  apprehension  easy, 
while  others  favour  analysis  or  attention  to  a  part  rather  than  to  the 
whole.  The  work  of  the  musical  artist  is  to  bring  these  two  attitudes 
under  control,  so  that  he  may  be  able  to  guide  the  hearer's  attention 
to  any  aspect  of  tone  he  pleases;  or  so  to  construct  his  tonal  masses 
that  listeners  on  the  average  will  tend,  with  a  minimal  deviation,  to 
devote  their  minds  to  those  aspects  of  tone  upon  which  the  artistic 
effect  has  been  built.  For  this  purpose  the  artist  must  know  as  much 
as  possible  which  factors  favour  each  attitude  and  what  power  each 
factor  has.  Consequently  the  science  of  music  is  called  upon  to  bring 
these  factors  into  the  fullest  light  of  knowledge  and  to  explain  as  exactly 
as  may  be  how  each  one  achieves  its  effect. 


CHAPTER  X 

THE  EQUIVALENCE  OF  OCTAVES 

The  equivalence  of  octaves  at  first  glance  seems  clearly  to  rest  primarily 
upon  the  fact  of  the  high  degree  of  fusion  appertaining  to  coincident 
tones  an  octave  apart.  For  that  is  the  ultimate  fact  of  tonal  hearing 
that  most  resembles  the  equivalence  of  octaves  in  music.  Such  pairs 
are  very  often  mistaken  for  a  single  tone,  more  often  than  happens 
with  any  other  interval.  When  octave  tones  are  sounded  in  succession, 
there  is  of  course  no  such  approximation  to  the  sound  of  a  single  tone, 
but  there  is  an  evident  connexion  between  the  two  which  reminds  us 
of  the  transition  from  a  thing  to  its  replica,  and  which  we  therefore 
incline  to  call  by  the  name  of  similarity  or  identity  or  equivalence  or 
the  like.  Which  of  these  terms  is  used,  depends  apparently  upon  the 
relative  importance  either  for  theoretical  or  for  practical  purposes 
that  is  ascribed  to  the  sameness  and  to  the  difference  of  the  two  tones. 
For  octave-tones  are  obviously  not  absolutely  the  same. 

But  although  the  primary  basis  of  the  equivalence  seems  so  obvious, 
the  system  of  facts  of  a  similar  nature  does  not  seem  to  confirm  it, 
at  least  in  practical  connexions.  For  the  octave  is  only  the  highest 
degree  of  a  series  of  grades  of  fusion  which  have  been  known  more  or 
less  satisfactorily  since  the  earliest  days  of  the  science  of  music.  This 
series  would  lead  us  to  expect  a  similar  grading  of  equivalence,  which 
by  no  manner  of  means  can  be  claimed  as  real.  We  cannot  call  the 
tones  of  a  fifth  similar  or  equivalent  as  we  call  those  of  the  octave, 
not  even  if  we  say  the  degree  of  similarity  or  equivalence  is  very  much 
less  than  in  the  octave.  It  is  true  that  crude  and  primitive  forms  of 
music  do  use  parallels  of  fifths  in  the  same  way  as  we  use  parallels  of 
octaves  in  our  music.  But  even  so  the  use  is  nothing  like  so  extended, 
nor  has  it  survived  the  first  refinements  of  musical  taste.  Fifths  are 
then  no  more  equivalent  than  fourths  or  thirds  or  seconds  are.  But 
the  equivalence  of  octaves  is  of  the  greatest  and  most  extended 
importance  in  all  music;  far  from  being  merely  a  primitive  crudity, 
it  increases  in  importance  with  the  development  of  music. 

Its  central  importance  for  musical  practice  and  theory  dates  from 
the  famous  doctrine  of  Jean  Philippe  Rameau  concerning  the  inversions 

W.  F.  M.  5 


66  THE  EQUIVALENCE  OF  OCTAVES  [ch. 

of  chords.  In  the  preface  to  his  simplification  of  Rameau's  teaching 
D'Alembert  pointed  out  that  up  till  then  work  "had  been  confined 
almost  completely  to  the  collection  of  rules  mthout  reasons  for  them; 
there  had  been  no  discovery  of  analogy  and  of  a  common  source; 
blind  trial  had  been  the  sole  compass  of  artists."  "M.  Rameau,"  he 
wrote,  "is  the  first  to  begin  to  dispel  this  fog  of  chaos.  In  the  resonance 
of  the  sonorous  body  he  has  found  the  most  probable  origin  of  harmony 
and  of  the  pleasure  it  causes  us  :  he  has  developed  this  principle  and 
shown  how  the  phenomena  of  music  emerge  from  it :  he  has  reduced 
all  the  chords  to  a  small  number  of  simple  and  fundamental  chords, 
of  which  the  others  are  only  combinations  and  inversions;  finally  he 
has  succeeded  in  apperceiving  the  mutual  dependence  of  melody  and 
harmony  and  in  making  it  felt"  (9,  vif.).  "Whatever  may  be  the 
fruit  of  the  further  efforts  of  others,  the  fame  of  the  learned  artist  has 
nothing  to  fear;  he  will  always  have  the  merit  of  having  been  the 
first  to  make  music  a  science  worthy  to  occupy  philosophers;  of  having 
simplified  and  facilitated  its  practice;  of  having  taught  musicians  to 
carry  into  this  region  the  torch  of  reasoning  and  of  analogy"  (9,  xviii). 
Later  on  (9,  222)  he  wrote  that  a  certain  special  difficulty  and  some 
others  less  considerable,  would  not  prevent  fundamental  basses  from 
being  "the  principle  of  harmony  and  of  melody;  as  the  system  of 
gravitation  is  the  principle  of  physical  astronomy,  although  this  system 
does  not  account  for  all  the  phenomena  that  are  observed  in  the  move- 
ment of  the  celestial  bodies."  The  idea  of  the  connexion  between 
chords  that  involve  the  same  notes  of  the  octave  is  so  familiar  to  the 
modern  musical  mind  that  it  is  necessary  to  recall  clearly  that  the 
idea  did  not  always  stand  in  the  forefront  of  the  musician's  cognitive 
consciousness.  He  may  always  have  felt  it,  to  be  sure,  but  he  certainly 
did  not  always  know  that  he  felt  it^  (cf.  60,  43). 

^  Readers  who  look  upon  the  connexion  of  inversions  as  perfectly  obvious  will  be 
interested  in  a  quotation  from  a  contemporary  of  Rameau's  to  whom  the  latter's  doctrine 
came  as  a  novelty : 

"We  must  not  omit  an  observation  most  easy  to  make  at  this  point  and  also  of  the 
greatest  moment  for  the  clearness  and  solidity  of  the  doctrine  we  have  been  gradually 
expounding.  A  concert  has  need  for  example  of  three  voices  if  it  is  to  embrace  with  their 
help  three  consonances,  prime,  third  and  fifth,  which  are  called  by  the  masters  Harmonic 
Triad.  The  prime  is  always  foimd  placed  in  the  lowest,  the  third  in  the  middle,  the  fifth 
in  the  highest  place.  Now  suppose  that  the  prime  is  moved  to  the  higher  octave,  so  that 
the  third  remains  in  the  lowest  place.  The  ear  is  no  longer  satisfied  with  it.  It  no  longer 
seems  that  the  concert  is  finished.  Hence  the  concord  does  not  feel  that  it  has  yet  returned 
to  that  note  whence  it  has  taken,  and  in  which  it  recognises,  its  origin  and  in  which  alone 
it  can  come  to  rest  and  finish.    It  is  openly  apparent  that  the  harmony  is  suspended. 


X]  THE  EQUIVALENCE  OF  OCTAVES  67 

Rameau  himself  actually  thought  that  we  fail  to  distinguish  octaves 
in  "the  resonance  of  the  sonorous  body."  The  partials  1,  2,  4,  8  and 
16  are,  of  course,  octaves,  which,  he  said,  really  resonate  even  more 
loudly  than  do  those  numbered  3  and  5,  because  of  the  size  of  the 
resonating  parts  of  the  musical  instrument.  So  even  though  we  fail 
to  distinguish  them,  we  are  nevertheless  necessarily  affected  by  them, 
"but  by  an  occult  feeling  that  has  so  far  prevented  us  from  discovering 
its  cause"  (54,  3).  From  this  feeling  our  sense  of  the  identity  of  octaves 
has  arisen.  We  actually  prefer  to  have  tones  closer  together  than  they 
are  offered  to  us  by  nature  in  the  series  of  partials,  in  order  that  we  may 
have  them  within  the  range  of  the  voice.  For  as  we  do  not  distinguish 
the  octaves  amongst  the  partials,  the  range  of  the  voice  is  soon  exceeded. 
Thus,  1 ,  3  and  5  take  us  up  through  a  range  of  two  octaves  and  a  third, 
and  they  actually  include  only  one  interval  less  than  the  octave,  namely 
the  major  sixth  between  3  and  5.  The  ear  also  finds  it  easier  to  move 
about  amongst  close  intervals  because  of  the  short  distance  between  their 
tones.    But  the  identity  of  octaves  does  not  prevent  them  from  intro- 

This  prime  voice  has  nevertheless  not  been  omitted.  We  have  done  naught  but  transfer 
it  from  the  lowest  place,  where  it  stood,  to  the  highest.  We  have  still  in  ear  the  same  three 
notes.  How  then  has  so  great  a  change  in  the  effect  of  all  been  made?  To  imagine  hearing 
the  low  octave  of  a  high  note  that  we  actually  hear  is  very  easy  for  anyone.  So  in  this 
way  we  shall  be  able  to  make  up  for  the  defect,  replacing  in  fantasy  the  true  bass  in  the 
place  whence  it  was  taken.  Then  we  shall  have  this  principal  voice  present  with  us  in 
two  places:  once  in  the  high  part  where  we  hear  it,  and  again  in  the  low  part  where  we 
imagine  it.  Our  ear  will  nevertheless  not  yet  be  satisfied.  We  shall  still  not  hear  the  perfect 
chord,  the  chord  that  concludes.  And  why  so?  Because  the  force  of  the  high  voice  that 
is  really  heard  prevails  over  the  force  of  the  low  voice  that  would  only  be  imagined. 
The  sense  of  the  ear  that  is  the  natural  judge  of  harmony,  does  not  let  itself  be  deceived 
by  the  imagination.  It  would  still  refer  the  two  prime  voices  that  are  actually  heard  to 
the  third  which  is  also  sensed;  and  thus  the  harmony  would  still  remain  imperfect.  It 
would  not  refer  it  to  the  imaginary  fourth  voice  by  reference  to  which  alone  the  two 
higher  notes  could  change  their  proportions  and  render  themselves  apt  to  conclude. 
This  most  simple  observation  which  turns  upon  an  experience  known  to  everyone  and 
beyond  all  doubt,  proves  that  the  common  statement  that  the  one  octave  is  the  equivalent 
of  the  other  requires  some  limitation.  In  a  large  number  of  cases  the  statement  is  true, 
but  not  in  all.  In  particular  it  is  always  false  in  reference  to  notes  that  do  duty  as  bass; 
the  which  in  changing  place  change  their  nature  and  make  the  nature  change  of  all  others 
from  below  which  they  withdraw. 

Now  if  that  is  so  (and  nobody  can  deny  it),  however  could  truth  or  at  least  verisimilitude 
belong  to  the  new  doctrine  of  inversions  which  is  nowadays  so  celebrated  as  a  thing 
most  useful  to  the  art  and  perhaps  the  most  noble  secret  that  has  yet  been  discovered  in 
harmony  ? 

To  me  there  seems  to  be  nothing  to  recognise  in  it  but  error  and  perversion"  (57,  sss.). 

It  is  clear  from  the  above  that  the  'sameness'  of  octaves  is  not  the  same  idea  as  the 
equivsdence  of  inversions  (of.  60,  37). 

6—2 


68  THE  EQUIVALENCE  OF  OCTAVES  [ch. 

ducing  some  differences  into  harmony  and  melody.  But  that,  Rameau 
said,  consists  "only  in  the  different  modifications  of  one  and  the  same 
whole  differently  combined,  where  sounds  cannot  change  their  order 
without  the  help  of  their  octaves"  (54,  13).  The  sounding  of  an  octave 
in  place  of  the  fundamental  in  no  way  distracts  the  ear  from  the  natural 
whole  that  guides  it;  the  ear  recognises  the  fundamental  sound  in  its 
octaves,  no  matter  what  the  order  of  the  parts  of  the  chord ;  it  is  always 
reminded  of  this  same  whole  (given  in  the  resonance  of  the  sonorous 
body).  If  the  chord  is  consonant,  it  is  equally  so  in  all  its  combinations. 
In  short,  2,  4,  8  and  1  are  for  us  but  one  sound,  in  which  1  always 
presides,  whether  we  hear  it  or  not  (54,  16),  Identity,  Rameau  added, 
may  seem  rather  an  extreme  term  to  use,  but  you  may  adopt  any 
term  you  like  so  long  as,  not  going  so  far,  it  goes  far  enough. 

That  is  precisely  the  difficulty  in  this  problem — to  find  a  theoretical 
basis  that  will  evidently  go  as  far  as  is  needful  in  establishing  sameness 
and  with  equal  evidence  refrain  from  obliterating  the  differences  that 
feeling  and  practice  demonstrate.  Rameau  certainly  overdid  the  aspect 
of  sameness.  He  admitted  himself  that  we  follow  in  our  music  the 
traces  given  by  nature  in  the  resonance  of  the  sonorous  body  "only 
by  the  grace  of  these  octaves"  (41,  36).  That  is  perfectly  plain;  it  has 
often  been  laid  as  a  primary  difficulty  against  those  who  claim  to  derive 
the  tones  of  the  scales  from  the  series  of  partials.  How  are  you  going 
to  bring  them  down  from  their  heights  to  within  the  range  of  an  octave? 
Some  second  principle  is  obviously  required  for  this  purpose.  This 
was  given  for  Rameau  in  his  "occult  feeling."  Without  that  the  needs 
of  the  voice  would  remain  unsatisfied;  or  rather  the  voice  would  have 
had  to  be  devised  so  as  to  cover  a  much  larger  range.  And  the  ear 
would  likewise  have  had  no  scope  for  preferences  as  to  the  sizes  of 
intervals. 

Rameau  was  also  right  in  claiming  that  in  the  different  inversions 
we  are  reminded  of  a  certain  whole,  but  that  whole  is  not  the  octaves 
I,  2,  4,  8  and  16.  Such  an  answer  would  be  elicited  from  the  mind  of 
no  musician  unlearned  in  the  claims  of  theory.  But  any  musician  would 
answer — more  or  less  so — that  the  whole  recalled  to  his  mind  is  the 
group  of  all  possible  combinations  of  the  chord.  When  asked  to  name 
a  given  one  of  them,  he  will  say  :  "that  is  (the — arrangement  of)  the — 
inversion  of  the — chord."  When  one  of  these  combinations  is  heard, 
the  common  relations  are  also  'in  some  way'  heard,  but  the  differences 
peculiar  to  it  are  equally  evident.    Though  the  parts  out  of  which  the 


X]  THE  EQUIVALENCE  OF  OCTAVES  69 

chord  is  composed  and  'in  some  manner'  the  whole  that  it  forms  are 
always  the  same,  yet  the  consonance  is  by  no  means  the  same,  nor  is 
the  harmonic  treatment,  although  it  may  well  be  the  case  that  inversion 
will  not  turn  any  consonance  into  a  dissonance,  or  conversely.  For  the 
purposes  of  science,  it  is  just  the  'some  way'  and  'some  manner'  that 
is  the  problem.  That  is  what  we  must  give  precise  form  to,  so  that  the 
practical  and  sensory  consequences  so  clear  to  the  musician  may  follow 
evidently  from  as  clear  a  conceptual  foundation. 

Helmholtz's  theory  has  seemed  to  many  to  be  a  great  improvement 
upon  Rameau's  or  even  to  have  finally  solved  the  problem.  In  the 
octave  we  hear  again  a  part — at  best  the  half — of  what  we  heard 
before — the  fundamental  and  its  own  special  series  of  partials.  A  most 
suggestive  and  winning  explanation,  very  hard  to  abandon  even  when 
it  has  been  disproved  on  other  grounds!  A  theory  is  always  seductive 
when  it  has  a  fair  and  clear  speech  for  every  phase  of  the  business,  for 
every  doubt  and  hesitation,  and  withal  so  cleverly  conceals  the  fact 
that  the  basis  of  explanation  has  only  been  assumed;  this  basis  is  not 
really  patent  and  clear,  as  it  is  in  the  analogous  cases  referred  to  for 
support — the  synthetic  similarity  of  faces;  it  is  only  'just  as  good.' 
What  Helmholtz  failed  to  show  was  why  partial  tones  ever  come  to 
form  a  fused  synthetic  whole.  And  it  is  difficult  for  most  folks  to  appre- 
ciate the  importance  of  this  omission.  The  explanations  which  flow 
from  the  assumptions  are  for  all  ordinary  cases  apparently  so  neat  and 
apt  that  more  could  hardly  be  desired.  Further  demands  and  criticism 
look  like  finical  pedantry. 

And  yet  Helmholtz  secured  his  whole  basis  of  explanation  by  mere 
analogy — one  of  the  kind  that  can  be  stated  so  plausibly  for  either  of 
two  opposite  ends.  For  if  in  the  octave  we  hear  again  a  part  of  what 
we  heard  before,  that  should  lead  us  in  the  course  of  time  to  distinguish 
the  first  and  second  partials  of  a  tone  as  different  primary  tones.  The 
progress  of  musical  practice  would  thus  bring  about  a  gradual  analysis 
of  timbre  into  its  ultimate  constituents.  Some  psychologists  believe 
that  the  world  begins  for  the  child  in  William  James's  words,  as  "one 
great  blooming  buzzing  confusion"  (27,  488).  But  we  know  that  it  soon 
clears  up  into  its  many  distinct  parts.  Why  should  it  not  be  so  with 
the  parts  of  musical  tones  or  of  ordinary  tones?  Separation  and  separate 
handling  should  here  also  lead  to  mental  distinction  and  abstraction. 
Who  is  to  hold  the  balance  between  the  tendency  to  confuse  the  parts 
of  a  whole  with  one  another  or  with  the  whole  and  the  tendency  to 


70  THE  EQUIVALENCE  OF  OCTAVES  [ch. 

distinguish  the  separable  parts  of  the  whole  from  one  another  or  from 
the  whole?  Helmholtz,  after  all,  gets  no  further  than  does  Rameau 
with  this  "occult  feeling."  In  fact  this  phrase  better  conforms  to  the 
results  of  Stumpf's  criticism  of  the  theories  of  consonance  given  by 
Helmholtz  and  others  and  to  the  suggestions  finally  made  by  Stumpf 
as  to  the  probable  basis  of  fusion  in  some  synergy  of  the  nervous 
system.  The  "  occult  feeling  that  has  so  far  prevented  us  from  discovering 
its  cause"  is  just  what  we  might  expect  from  'synergy,'  which  is  an 
occult  (cerebral)  process  that  has  so  far  prevented  us  from  formulating 
its  nature. 

In  recent  years  an  attempt  has  been  made  to  account  for  the  equi- 
valence of  octaves  by  setting  up  the  series  of  differences  that  lie  within 
the  range  of  an  octave,  no  matter  what  its  general  pitch  may  be,  as 
the  primary  qualities  of  hearing.  Then  the  series  from  c  to  c',  whether 
it  be  taken  continuously  or  discretely  as  in  any  specific  scale,  is  a  series 
of  qualities  like  that  of  the  spectral  colours;  only  in  the  tones  we  do 
not  merely  just  return  to  the  starting-point  but  we  are  able  to  repeat 
the  series  a  number  of  times. 

This  theory  can  hardly  be  discussed  without  close  study  of  the 
psychological  notion  of  quality.  That  includes  all  kinds  of  sensation 
such  as  touch,  cold,  warmth,  pain,  sweet,  sour,  the  various  smells,  the 
colours  such  as  blue,  red,  etc.,  muscular  feeling,  hunger,  thirst,  etc.,  etc., 
all  as  specific  feelings,  without  concern  for  their  intensity  or  localisation 
or  for  any  other  distinguishable  aspects  of  them,  except  merely  their 
kind.  This  bare  kind  or  quality  is  the  thing  of  all  things  that  we  know 
perhaps  least  of  in  itself.  We  seem  to  have  some  understanding  of  it 
in  vision;  but  our  understanding  is  here  almost  solely  physiological. 
None  of  us  knows  what  inner  connexion,  if  any,  there  is  between  blue 
and  red,  or  between  yellow  and  blue,  as  felt  colours.  There  seems  to 
be  none,  and  yet  we  at  once  recognise  them  all  as  colours. 

One  of  the  characteristic  features  of  colours  is  their  changes  of  kind. 
Red  passes  through  orange,  that  resembles  it,  to  yellow,  that  is  like 
orange,  but  not  at  all  like  red.  A  similar  change  brings  us  to  green, 
then  to  blue,  and  finally  back  to  red  through  purple.  If  we  are  to  consider 
the  differences  included  within  the  octave  as  qualitative,  comparison 
with  colour  would  incline  us  to  look  for  characteristic  turning  points, 
as  it  were,  within  the  octave.  These  might  be  supposed  to  occur  at 
the  thirds,  fourth,  fifth  and  sixths  perhaps.  Various  suggestions  have 
been  made.    But  none  of  them  really  explains  the  peculiarities  that 


X]  THE  EQUIVALENCE  OF  OCTAVES  71 

would  be  thus  described^.  It  is,  of,  course,  conceivable  that  from  a 
detailed  study  of  classifications  made  in  relation  to  the  various  forms 
of  tone-deafness,  etc.,  a  good  and  probable  physiological  theory  of 
tonal  quality  might  in  time  be  obtained,  just  as  has  been  done  in  vision. 
Such  a  theory  might  then  explain  the  peculiar  relations  that  characterise 
thirds,  fourth,  fifth,  and  the  rest.  No  very  satisfactory  explanation 
has  as  yet,  however,  been  given  even  of  colour  afiinities.  The  prospects 
of  raising  a  lucid  theory  of  music  on  this  qualitative  basis  are,  to  say 
the  least,  not  yet  exciting. 

Of  course,  that  would  be  of  no  consequence  at  all  if  the  classification 
as  quality  were  logically  inevitable.  It  is  not  so  by  any  means.  Some 
objections  may  be  raised  to  the  theory  from  the  special  difficulties  it 
creates,  from  the  obscurity  of  the  ground  it  rests  upon,  and  from  the 
special  phenomena  of  quality  which  the  classification  must  introduce 
(cf.  77,  44  ft.).  But  until  exclusion  makes  one  theory  or  another  logically 
inevitable,  the  merits  of  theories  rest  upon  their  respective  powers  of 
accounting  for  all  the  facts.  The  theory  of  octave  qualities  does  not 
reduce  the  amount  to  be  explained.  The  assumptions  it  makes  require 
as  much  explanation  and  justification  as  do  the  facts  they  are  supposed 
to  explain.   And  a  better  explanation  can  be  given  without  them. 

The  intervals  in  common  use  and  their  inversions  are  reducible 
to  six  pairs: 

2,  II, 3,  III, 4, T 

VII,  7, VI,  6, 5, T. 

The  only  marked  change  in  grade  of  consonance  produced  by  inversion 
is  found  in  4 — 5 — i.  The  fifth  is  clearly  more  consonant  or  fused  than 
the  fourth.  But  in  all  but  the  tritone  a  decided  difference  is  wrought 
in  the  interval  itself.   In  the  volumic  theory  of  tone  already  developed 

*  As  the  octave  according  to  the  volumic  theory  is  the  greatest  approximation  towards 

the  balance  of  a  single  tone  that  two  simultaneous  tones  can  make,  and  the  fifth  is  the 

next,  we  might  expect  a  certain  parallelism  in  the  character  of  the  steps  by  which  we 

pass  from  the  two  ends  of  the  octave  to  the  fifth : 

0,  VII,  7,  VI,  6,     5\t,  ., 

^       '    „       '     .   >  Tntone 
p,     2,    II,   3,  III,  4/ 

Here  the  fourth  is  taken  as  the  counterpart  of  the  fifth,  as  it  were.  Otherwise  the  parallel 

will  only  hold  if  the  fourth  is  slumped  with  the  thirds  and  the  tritone  is  set  over  against 

the  minor  sixth  when  it  functions  (in  equal  temperament)  as  a  discord  (augmented  fifth). 

Thus: 

o,  VII,  7,      VI,  6,     6»,  6 

p,    2,    II,  3,  HI,  4,  T.  5. 
Cf.  Chap.  XXI  below. 


72  THE  EQUIVALENCE  OF  OCTAVES  [ch. 

we  have  good  ground  for  the  understanding  of  both  these  facts.  The 
balance  and  symmetry  of  very  different  intervals  may  be  approximately 
equal.  But  the  intervals  themselves  are  so  different  because  they  are 
quite  different  proportions  of  volumes. 

Now  the  musical  ear  is  not  restricted  to  a  knowledge  cf  the  simplest 
intervals.  These  are  naturally  of  great  importance  in  music,  because 
they  are  amongst  the  simplest  complexes  of  form  known  to  the  art. 
The  simplest  of  all  is  the  absolutely  pure  tone.  A  variant  upon  this  is 
the  blend,  which  gives  the  tone  a  surface,  as  it  were.  Interval  is  the 
first  step  that  involves  a  variable  proportion,  constant  only  for  each 
specific  interval.  But  it  is  only  the  first  step  on  a  long  Une  of  possible 
complications,  each  of  which  may  equally  well  be  learnt  as  a  definite 
complex  of  proportions,  or  as  a  'pattern.'  Let  us  follow  out  this  process 
of  complication,  beginning  with  the  addition  to  a  simple  interval  of 
the  octave  of  its  lower  tone.  The  'chord'  c,  e,  c^,  for  example,  may  be 
represented  thus  (Fig.  3)  : 


Fig.  3 

The  length  of  the  lines  represents  the  relative  length  of  the  volume 
of  each  tone  and  the  middle  points  mark  their  pitches — the  points 
that  predominate  in  each  volume  and  so  give  the  whole  a  definite 
mark  by  which  it  can  be  placed  in  the  series  of  all  the  tones.  The  volume 
of  a  tone  is,  of  course,  not  homogeneous  throughout  its  length,  as  the 
simple  line  suggests.  It  probably  varies  from  the  central  pitch-point 
towards  either  end  by  a  regular  decrease  of  intensity.  In  any  case  this 
variation  is  quite  regular,  so  that  each  tone  may  be  a  symmetrical  whole. 
The  range  of  the  variation — from  the  pitch  maximum  to  the  opposite 
ends  of  the  volume's  length — will  probably  be  the  greater,  the  louder 
the  tone  is.  Consequently  when  several  tones  overlap  to  form  a  chord, 
the  intensity  at  each  part  of  the  chord's  volume  will  vary  infinitely 
according  to  the  relative  strength  of  the  component  tones.  For  the 
overlapping  will  give  some  sort  of  summation  of  the  strength  of  each 
particle  of  sound  that  is  common  to  two  or  more  of  the  component 
tones  of  the  chord.  We  cannot  yet  say  precisely  what  the  mode  of  this 
summation  is.  However,  there  is  a  feature  of  every  chord  that  is  in 
no  way  affected  thereby,  namely  the  relative  or  proportional  position 
in  the  whole  volume  of  the  points  where  a  departure  from  the  regular 


X]  THE  EQUIVALENCE  OF  OCTAVES  73 

changes  that  constitute  the  balance  and  regularity  of  a  single  tone 
occurs.  And  these  points  are  bound  into  definite  sets  by  their  dependence 
on  the  physical  stimulus  of  sound.  They  are  always  the  same  for  any 
one  ratio  of  vibration.  And  they  are  psychically  fixed  by  the  ordinal 
character  of  the  points  themselves.  Thus  the  whole  volume  will  be 
marked  out  into  a  set  of  proportional  parts  properly  indicated  in  the 
diagram  given  above. 

The  musician  in  the  course  of  his  practice  is  made  thoroughly 
familiar  with  the  complex  cec^  both  as  a  whole  and  in  its  parts,  c,  e 
and  c^,  and  their  binary  combinations,  ce,  ec^,  cc^.  In  time  he  becomes 
able  to  survey  these  parts  within  the  whole  and  to  recognise  their 
presence,  either  in  an  absolute  way  by  naming  their  exact  pitches,  or 
in  a  relative  proportional  way  by  recognising  the  intervals  they  form. 
Even  though  he  cannot  banish  c  from  the  whole  complex,  he  can  survey 
the  '  upper '  parts  around  the  pitch-points  of  c^  and  e  and  recognise  the 
proportions  of  these.  Or  he  may  think  of,  and  attend  to,  cc^  and  recognise 
its  presence,  ignoring  e  the  while.  Or  he  may  dwell  upon  ce  and  ignore  c^ 
or  at  least  its  predominant  parts  about  its  pitch-point.  This  process 
of  abstraction  is  already  familiar  in  all  those  other  senses  which  show 
a  definite  field  or  an  ordinal  system,  such  as  touch  and  vision.  We  can 
shift  the  attention  easily  from  finger  to  finger  so  long  as  touch  sensations 
appear  in  either.  In  vision  we  are  much  more  expert  at  such  spatial 
or  ordinal  abstraction.  The  patterns  of  wall-paper  often  allow  of  com- 
bination and  recombination  in  the  most  varied  way. 

The  special  peculiarity  of  hearing  in  this  respect  is  the  relative 
slowness  with  which  the  average  person  acquires  practice  and  skill  in 
recognising  tonal  proportions  and  in  abstracting  them  from  complexes 
of  tones.  In  the  visual  field  we  can  move  patterns  from  place  to  place 
or  rotate  them,  and  dissect  them  as  we  please.  In  hearing  rotation  is 
impossible;  movement  within  the  sensory  field  is  only  possible  in  so 
far  as  pitch  and  volume  of  tone  are  altered  in  the  way  laid  down  by 
the  physical  stimulus;  and  dissection  is  limited  in  the  same  way.  These 
restrictions  make  analysis  so  hard  that  most  people  are  discouraged 
by  them.  But  they  are  easily  enough  overcome  by  those  in  whom  a 
good  ear  has  created  special  interest  and  enthusiasm. 

As  to  the  way  in  which  we  may  judge  of  the  fusion  of  parts  in  the 
whole  by  abstraction  from  the  whole,  there  is  some  difference  of  opinion. 
It  has  been  urged  that  fusion — the  degree  to  which  a  tonal  mass  appears 
to  resemble  the  unity  of  a  single  tone — must  necessarily  be,  and  is, 


74  THE  EQUIVALENCE  OF  OCTAVES  [ch. 

modified  by  the  addition  of  a  third  tone  to  any  pair.  The  mode  of 
alteration  will  depend  on  whether  the  new  tone  forms  a  greater  or 
less  fusion  with  either  of  the  two  tones  than  they  form  with  one  another. 
Yet  one  might  have  expected  the  united  fusion  always  to  be  worse, 
since  the  new  tone  necessarily  forms  a  lesser  degree  of  unity  with  either 
of  the  first  two  than  that  one  formed  by  itself,  being  a  tone,  i.e.  an  optimal 
unity.  Therefore  when  this  deteriorated  tone  is  added  to  the  other 
one  of  the  pair  first  given,  the  triad  resulting  should  always  be  less 
fused  than  the  original  pair.  Probably  a  good  deal  depends  upon  the 
point  of  view.  If  we  look  for  mere  plurality  of  tones,  any  trio  will  be 
more  plural  than  the  duo.  If  we  look  for  the  amount  of  good  balance 
or  fusion,  as  that  is  known  in  the  octave,  fifth,  etc.,  we  shall  find  more 
of  it  present  in  cgh  than  in  cb.  Here  mere  interpenetration  over  the 
whole  tonal  mass  is  not  so  much  the  standard,  as  perhaps  a  certain 
kind  of  interpenetration  already  familiar  in  various  forms.  There  may 
be  some  abstraction  in  the  process — i.e.,  a  local  abstraction  within  the 
ordinal  field  of  sound  of  the  chords.  It  is  evidently  not  easy  for  those 
who  experiment  upon  the  fusion  of  more  than  two  tones  to  make  their 
point  of  view  in  observation  quite  clear.  The  opposite  view  has  been 
upheld — that  the  addition  of  further  tones  makes  no  difference  whatever 
to  a  fusion  already  given.  This  seems  quite  a  reasonable  position 
provided  the  above-mentioned  'local'  abstraction  of  fusion  has  become 
easy  enough. 

There  seems  to  be  no  reason  in  the  nature  of  tonal  complexes  them- 
selves why  such  abstraction  should  not  succeed  with  those  who  are 
highly  gifted  and  practised  acoustically.  They  would  then  isolate  for 
attention  the  tones  in  question  and  see  the  sort  of  balance  and  symmetry 
they  possess.  Of  course  they  cannot  lift  the  tones  they  abstract  out  of 
the  whole  complex  they  are  abstracted  from.  Abstraction  here  means 
only  devoting  special  attention  to  certain  tones  and  recognising  in  them 
features  that  are  usually  characteristic  of  them  in  isolation,  in  so  far 
as  these  features  have  been  only  partially  or  not  essentially  distorted 
by  the  presence  of  the  other  tones.  Thus  one  who  abstracts  cc^  from 
the  chord  cec^  will  notice  that  the  maximal  volume  of  the  chord  is  that 
of  c;  that  c^  is  present  as  a  pitch  at  its  usual  ordinal  place,  that  the 
parts  of  the  tone  lying  around  the  pitch-point  have  the  proper  tonal 
symmetry  and  that  there  is  the  usual  clean  function  at  the  pitch-point 
of  c.  Of  course  the  tone  e  will  often  be  encountered  during  this  process. 
But  one  who  is  highly  practised  may  pass  to  and  fro  about  this  irrelevant 
tone  without  being  disconcerted  by  it,  and  may  feel  as  able  to  give 


X]  THE  EQUIVALENCE  OF  OCTAVES  75 

his  judgment  as  he  would  if  it  were  not  there.  For  others,  however, 
the  third  tone  may  be  a  source  of  great  disturbance  and  they  may  feel 
they  never  really  can  ignore  it,  so  that  it  always  spoils  the  effects  for 
them. 

Later  on  we  shall  meet  evidence  that  will  call  for  a  more  special 
effort  to  settle  the  question  of  the  part  played  by  the  fusion  of  single 
intervals  in  chords. 

The  '  chord '  cec?^  is  in  the  experience  of  the  musician  not  only  given 
at  all  levels  of  pitch,  but  when  it  occurs  at  any  pitch,  it  then  commonly 
occurs  at  the  octaves  above  and  below.  Thus  cec^  may  be  carried  up 
and  down  over  the  piano,  cec^e^(^e^c^  (Fig.  4).  The  new  parts  here  make 
no  significant  change  in  the  diagram  of  volumes,  e^  fits  in  between  the 
pitch  of  e  and  the  common  upper  limiting  point  of  all  tones,  while  c^ 


Fig.  4.   ninstrating  the  continuity  of  'pattern'  made  possible  by  the  volumic  relations  of 
the  octave  and  upon  which  the  connexions  of  inversions  of  the  '  same '  chord  rest. 

likewise  fits  in  between  the  pitch  of  c^  and  that  point.  Both  of  these 
latter  tones  can  be  taken  as  mere  appendages  of  their  lower  octaves, 
the  more  so  the  weaker  their  strength  is.  In  any  case  the  new  tones  do 
not  spoil  the  previous  pattern,  but  merely  continue  it  further  towards 
the  upper  limit  of  hearing  in  the  same  characteristically  proportionate 
form.  The  component  tones  of  the  whole  are  not  more  easily  separable 
for  their  being  so  many  but  less  so.  Only,  the  characteristic  pattern 
of  the  whole  remains  the  same  and  can  be  recognised  with  almost,  if 
not  quite,  equal  ease. 

Now  if  ecV  or  its  extensions  are  given  in  the  same  way,  they  may 
not  only  be  analysed  into  the  same  musical  components  c  and  e,  but 
they  give  a  pattern  which  is  partly  the  same  as  that  of  cec^.  The  closeness 
of  the  resemblance  is  the  greater,  of  course,  the  further  up  the  pattern 
is  extended,  ec^e^c^e^,  etc.  But  it  is  obvious  that  cec^  and  ec^e^  are 
identical  in  respect  of  their  common  part  ec*.  This  part  will  make 
them  thus  far  similar.  And  the  resemblance  is  increased  by  the  similar 
way  in  which  the  other  tone  is  related  to  one  of  the  tones  of  the  common 
pair.  Of  course  this  sort  of  similarity  is  evident  both  in  the  mere  musical 
symbols  and  their  arrangement  and  in  the  common  musical  consciousness 


76  THE  EQUIVALENCE  OF  OCTAVES  [ch. 

of  our  time.  Our  concern  here  is  to  show  definitely  how  this  similarity 
is  grounded  upon  the  sensory  material  of  hearing,  in  the  tones  themselves. 
The  similarity  that  appears  to  the  musical  mind  is  therefore  not  merely 
the  result  of  musical  analysis  or  of  theory  or  of  thought,  but  is  a  true 
representation  of  the  relations  of  the  parts  of  the  sensory  stuff  of  music. 

Thus  we  do  right  to  consider  cec^  and  ec^e^  in  a  certain  respect  as 
mere  aspects  of  one  another,  or  to  consider  ec^e^  as  a  trifling  alteration 
of  cec^,  which  for  some  reason  we  look  upon  as  the  normal  or  more 
fundamental  form.  The  same  holds  for  any  other  chord,  no  matter 
how  complex  and  discordant,  ceg  in  a  certain  respect  appears  again 
in  egc^  and  in  gc^e^.  They  are  all  patterns  that  may  be  said  to  be  parts 
of  their  common  extension  cegc^e^g^c^,  etc.,  except  that  the  common 
pattern  does  not  begin  at  the  same  part  of  its  cycle,  so  to  speak. 

The  musical  consciousness  that  has  got  thus  far,  will  find  it  easy 
to  see  the  same  pattern  even  when  its  parts  are  scattered  more  widely 
through  the  octaves  of  its  extension.  Thus  we  come  to  forms  such  as 
cge^,  ce^g^,  ge^(^,  the  familiar  positions  of  the  various  inversions.  The 
connexion  of  these  with  the  fundamental  pattern  ceg  cannot  remain 
obscure  after  the  musical  mind  has  learned  so  much  as  to  be  able  to 
create  ceg  itself,  to  know  it  and  to  use  it.  Of  course  this  pattern  is  only 
relatively  feebly  indicated  in  cg^e^ ;  but  for  the  practised  musical  mind — 
and  we  are  here  dealing  only  with  practice  in  the  simplest  things, 
though  for  the  foundations  of  the  science  they  are  the  hardest  problems — ■ 
the  connexion  is  as  plain  as  daylight,  as  plain  as  is  the  ordinary  hand- 
writing of  our  own  language  in  spite  of  its  great  variations  from  the 
copperplate  model. 

In  ordinary  music,  moreover,  there  is  a  much  greater  resemblance 
between  cg^e^  and  ceg  than  appears  in  the  great  intervals  between  the 
parts  of  the  former,  or  in  the  diagrammatic  representation  of  it  on  the 
basis  of  absolutely  pure  component  tones.  For  these  fundamental 
tones  are  ordinarily  accompanied  by  upper  partials.  If  we  suppose 
merely  that  the  lower  partials  are  present  in  some  strength,  we  should 
get 


c 

g^ 

e2 

c     ci     gi     c2     e2     g2   (b2t>)    c3   (d^)   e^    g3 

gl                          g2                             (d2)               g3 

e2                                   e3 

(b3)  (d*)          g* 

(b3)        e*          g*if 

Total 

c    c*    g^    c2    e2    g2               c3          e3    g3 

e*    g* 

In  this  series  the  pattern  ceg  is  represented  more  than  twice.    The 
resemblance  would  therefore,  be  more  evident  in  instrumental  tones 


X]  !  THE  EQUIVALENCE  OF  OCTAVES  77 

than  in  pure  tones.  The  musician  here  is  not  expected  to  analyse 
partials — a  thing  he  rarely  does  at  all.  The  point  is  only  that  these 
partials  will  reinforce  an  effect  that  is  apparent  enough  to  him  already 
on  common  psychological  grounds  without  partials.  Partials  alone 
would  not  suffice,  without  the  volumic  basis.  But  granted  that,  they 
will  only  repeat  and  confirm  it.  The  same  holds  true  to  some  extent 
for  the  difference-tones.  Thus  between  tones  whose  vibrations  stand 
in  the  ratio  of  1  :  3  (e.g.  c  and  g^),  the  first  difference- tone  (h-l)  will 
have  the  ratio  2,  and  so  will  form — even  in  the  case  of  pure  tones — a 
link  towards  the  filUng  out  of  a  pattern  and  the  extension  of  connexions 
by  fusion  beyond  the  octave. 

But  in  all  this  we  must  not  omit  to  notice  that  there  are  marked 
differences  between  the  different  inversions  and  their  different  positions. 
Therefore  we  observed  above  that  these  forms  were  identical  'in  a 
certain  respect.'  Apart  from  that  and  for  other  purposes,  their  differences 
are  great;  and  naturally  too.  The  bass  is,  as  above  explained,  the 
weightiest  part  of  the  chord,  its  centre  of  gravity  so  to  speak;  and  it 
must  make  a  great  difference  which  part  of  the  basal  pattern  bears 
this  function.  That  is  perfectly  familiar  in  musical  practice  and  theory, 
and  just  as  clear  on  our  theory.  The  pattern  set  by  gc^e^  continues  as 
g^c^e^,  etc.  It  has  the  same  series  as  ceg,  once  it  is  well  started;  but, 
as  given,  it  designates  the  gc^e^  complex  unit  of  pattern  most  strongly. 
If  we  care  to  ignore  that  designation  or  are  specially  led  to  do  so,  then 
we  may  well  see  the  ceg  type  most  of  all.  What  the  special  differences 
between  these  types  of  the  same  basal  pattern,  as  it  were,  consist  in 
essentially,  we  shall  endeavour  to  show  as  we  proceed. 

We  have  thus  shown  that  the  equivalence  of  octaves  rests  upon  a 
sufficient  natural  basis  in  the  stuff  of  tones  themselves.  And  our  account 
in  no  way  inclines  us  to  underrate  the  differences  that  exist  in  that 
sensory  stuff  between  the  different  groupings  of  tones  that  are  equi- 
valent. There  is  only  equivalence  for  certain  purposes.  A  point  of 
view,  an  attitude,  or  a  certain  trend  of  abstraction  has  to  be  made 
for  the  equivalence  to  emerge  so  strongly  as  to  suggest  sameness. 
Other  attitudes  may  concentrate  in  other  ways,  and  see  practically 
nothing  but  difference.  For  certain  purposes  there  is  familiarly  a  very 
considerable  difference  between  ceg  or  egc^  and  gch^. 

The  equivalence  thus  established  for  octaves  does  not  apply  to  any 
other  interval  in  the  same  way.   Suppose  we  double  the  interval  of  the 


78  THE  EQUIVALENCE  OF  OCTAVES  [ch. 

fifth — cgcP^  (Fig.  5),  The  second  fifth  does  not  fit  into  the  first  so  as 
to  be  a  mere  repetition  of  its  pattern.  It  would,  no  doubt,  do  so,  if 
each  tone  consisted  only  of  the  half  of  its  actual  volume  that  lies  on 
the  upper  side  of  its  pitch  {u  in  the  diagram  and  not  I).  But  these  I 
parts  break  into  one  another  irregularly.  The  l-end  of  the  second  fifth 
strikes  in  between  the  pitch-points  of  the  two  lower  fifths.  In  the  case 
of  the  octave  the  I  parts  of  the  higher  tones  merely  repeat  or  emphasise 
the  pitch-points  of  their  lower  octaves,  so  that  no  new  or  disturbing 
element  is  introduced.  The  higher  tones  merely  carry  onwards  and 
upwards  the  pattern  already  given  by  the  simple  interval  or  chord. 
Of  course  we  can  attend  to  the  one  or  other  fifth  in  the  whole  and 
hear  it  as  a  fusion,  but  the  two  do  not  follow  upon,  or  fit  into,  one 
another  as  a  continuation  of  one  whole  pattern.  If  gd^  is  played  after 
eg,  and  the  attention  is  concerned  with  their  justness  as  fifths,  gcP^ 
will  be  heard  as  the  repetition  of  eg.  Or  if  the  attention  is  concerned 
with  the  indirect  relation  holding  between  c  and  #  through  a  real  or 
imaginary  g,  it  will  take  a  similar  attitude.    But  if  gd^  follows  c^  as  a 

"        I  i         «  "  •^- 

•  -9 

• Ti C 


I  '  « 

Fig.  5 

part  of  a  whole  to  which  both  belong,  the  attention  directed  to  this 
pattern  will  not  find  itself  rewarded.  This  shows  more  clearly  than 
does  the  octave  that  equivalence  of  octaves  is  not  the  mere  repetition 
of  a  tone  of  the  same  'quality'  absolutely  inherent  in  itself,  or  of  the 
same  single  interval  of  two  tones;  but  it  is  the  presence  of  a  pattern 
of  which  a  chord  of  any  number  of  tones  forms  a  part,  or  the  indication 
of  that  pattern  in  a  way  that  suffices  for  the  musical  ear  under  the 
circumstances  of  the  moment. 

As  the  fifth  is  thus  distinguished  from  the  octave,  so  are  the  other 
intervals.  The  octave  is  the  only  interval  that  thus  makes  possible 
the  extension  of  patterns.  And  it  does  so  because  it  packs  the  repetition 
entirely  into  the  upper  half  of  its  lower  tone,  making  only  one  new  point 
of  predominance  at  the  upper  pitch-point.  That  is  why  the  octave 
is  of  such  fundamental  regulative  importance  in  all  music,  and  why 
the  equivalence  of  the  octave,  instead  of  being  a  survival  from  primitive 
forms  of  music,  is  of  constantly  increasing  importance. 

The  explanation  we  have  given  also  shows  that  the  equivalence 
of  octaves  does  not  rest  entirely  on  their  great  fusion,  as  such.  Of  course 


X]  THE  EQUIVALENCE  OF  OCTAVES  79 

the  reason  for  the  great  fusion  is  closely  allied  to  that  which  makes 
equivalence  possible.  But  the  equivalence  does  not  rest  on  the  balance 
of  proportions.  For  if  it  did,  the  fifth  as  already  noticed,  would  provide 
an  equivalence  of  second  grade,  which  is  not  really  found.  The  fifth 
holds  a  steady  second  place  in  music  to  the  octave  only  as  a  fusion, 
i.e.  as  a  consonance  in  our  music,  or  as  a  form  of  homophony  in  primitive 
music.  Equivalence  rests,  not  upon  the  balance  of  the  parts  of  a  single 
interval,  but  upon  the  way  in  which  octaves  extend  the  pattern  of 
proportions  of  an  interval  or  of  a  chord  without  distorting  that  pattern. 
Equivalence  thus  introduces  an  important  new  form  into  musical 
structure.  This  form  is  present,  indeed,  in  mice  in  the  single  interval, 
but  it  only  emerges  clearly  as  an  important  specialty  when  chords  are 
freely  used,  and  when  they  have  been  for  some  time  steadily  apprehended 
for  the  purposes  of  musical  structure  in  a  special  way,  i.e.  not  as  fusions, 
but  in  another  way,  which  we  have  classified  as  'pattern.'  Thus  we 
can  now  well  understand  why  the  notion  of  the  equivalence  of  inversions 
did  not  take  clear  shape  in  the  musical  mind  until  the  period  of  harmonic 
music  had  been  fully  inaugurated  and  had  had  time  to  ripen  into 
conscious  formulation  in  Rameau  or  his  more  immediate  precursors. 

In  other  words  that  make  it  almost  a  truism,  a  system  of  connexions 
like  those  of  inversion  is  only  possible  when  the  terms  connected  have 
become  familiar.  Simple  though  this  is,  it  is  so  important  that  it  may 
be  set  up  almost  as  a  principle  for  the  study  of  chords  in  so  far  as  the 
notion  of  inversion  reduces  these  to  a  manageable  number.  We  can 
speak  of  inversion  properly  only  when  we  know  that  the  best  (average) 
listeners  are  so  familiar  with  the  different  chords  as  to  be  able  to  recognise 
them  readily  as  parts  of  the  same  pattern.  After  all  any  set  of  notes 
whatever  can  by  suitable  (octaval)  transposition  be  inverted  into  a 
series  of  major  or  minor  thirds  with  appropriate  omissions.  No  real 
system  of  chords  can  be  founded  on  such  merely  formal  considerations. 
The  primary  factual  question  for  every  system  of  chords  that  uses  the 
notion  of  inversion  is  :  are  the  inversions  recognised  by  direct  hearing 
as  parts  of  one  pattern  ?  Contrariwise,  the  formal  reduction  of  all  chords 
to  columns  of  thirds  does  nothing  at  all  to  prove  either  the  real 
importance  or  the  real  primacy  of  the  third  in  musical  structure. 

The  preceding  exposition  cannot,  of  course,  prejudice  the  efforts 
of  abstraction  in  listening  to  chords  that  may  still  become  possible 
to  the  musical  mind.  It  is  conceivable,  for  example,  that  a  mind  might 
contrive  to  attend  to  a  column  of  simultaneous  fifths  or  of  any  other 


80  THE  EQUIVALENCE  OF  OCTAVES  [ch.  x 

interval  apart  from  their  mutual  interference  and  blurring.  Great 
concentration  and  practice  in  distinguishing  pitches — rather  than 
intervals — might  lead  this  way.  Those  who  have  absolute  ear  often 
recognise  intervals  rather  by  inference  from  their  absolute  pitches 
than  by  direct  apprehension  of  intervallic  proportion.  So  a  mind 
might  come  to  hear  chords  as  columns  of  pitches  standing  at  propor- 
tionate distances  from  one  another  rather  than  as  volumic  patterns 
of  proportional  nature  throughout.  Of  course  much  that  is  of  the 
greatest  value,  if  not  essential,  to  music,  would  thereby  be  abandoned — 
all  harmonic  effect  in  particular.  Perhaps  some  of  the  latest  experiments 
in  music-making  tend  in  this  direction;  for  example  Scriabin's  columns 
of  'fourths.'  Only  the  further  developments  of  this  line  of  construction 
and  the  general  judgment  passed  upon  it  in  the  course  of  time  will  show 
whether  it  has  struck  upon  new  and  useful  faculties  of  musical  analysis 
that  will  serve  the  synthetic  ends  of  artistic  creation.  If  we  cannot 
discover  whether  Scriabin  had  a  special  attitude  of  listening  to  his 
own  music,  we  shall  have  to  see  whether  in  time  such  an  attitude  will 
not  prove  to  be  essential  for  the  artistic  apprehension  of  his  works. 


CHAPTER  XI 

CONSECUTIVE  FIFTHS 

The  rule  forbidding  consecutive  fifths  is  one  of  the  fundamental 
generalisations  of  musical  structure.  The  view  is  indeed  sometimes 
expressed  that  modern  developments  have  swept  all  the  rules  of  harmony 
away,  and  that  this  one  like  others  no  longer  holds  because  composers 
break  it  repeatedly.  It  is  true  that  the  rule  has  its  exceptions.  But 
the  special  means  required  to  make  such  exceptions  tolerable  and  their 
late  appearance  in  any  frequency  in  the  highly  developed  art  show  that 
the  rule  has  really  the  fundamental  importance  commonly  ascribed 
to  it.  The  breaking  of  an  established  rule  is  naturally  the  first  fact 
to  engage  the  attention,  when  it  has  been  broken.  The  next  question 
inevitable  for  a  mind  that  feels  the  good  effect  produced  in  spite  of  the 
breach  of  rule  is  :  what  other  elements  of  the  whole  in  which  the  fifths 
appear,  are  responsible  for  the  good  effect?  The  pleasantness  of  fifths 
in  a  certain  setting  by  no  means  discredits  their  prohibition  under 
most  circumstances.  This  could  be  gainsaid  only  by  the  pedant  who 
lives  on  rules  and  does  not  apprehend  the  structures  he  studies  in  their 
primary  aspect — aesthetically — at  all.  But  the  sole  standard  of  art 
is  the  beauty  inherent  in  the  created  object.  We  do  right  to  expect 
art  to  be,  like  nature,  a  realm  of  law  and  order,  not  the  sport  of  chaotic 
chances ;  and  the  study  of  its  laws  is  the  science  of  art.  Rules  are  merely 
the  expressions  of  the  probable  sequences  of  cause  and  effect  already 
recognised.  They  are  useful  because  in  many  cases  they  foretell  the 
effect  with  accuracy.  But  if  their  prophecy  is  false,  they  must  be 
corrected  by  a  further  study  of  the  new  effects,  under  the  assumption 
that  the  effect  is  not  the  result  of  the  one  cause  stated  in  the  rule,  but 
is  the  resultant  of  a  number  of  causes,  some  of  which  act  in  opposition 
to  one  another  and  so  produce  from  time  to  time  apparently  contrary 
effects. 

The  prohibition  of  consecutive  fifths  appeared  comparatively  early 
in  the  history  of  the  art,  much  earlier  for  example  than  the  formulation 
of  the  connexions  of  inversions.  In  the  music  of  the  ancient  Greeks, 
series  of  fifths  or  of  fourths  seem  to  have  been  freely  allowed  in  instru- 
mental, but  not  in  vocal  music.    This  was  known  as  the  'antiphonic' 


82  CONSECUTIVE  FIFTHS  [ch. 

style.  In  vocal  music  only  octaves  were  run  in  series;  no  other  con- 
sonance was  ' magadised '  (16,  21, 154  ff.).  On  the  common  instruments 
of  Greek  music,  the  lyre  and  the  cither,  sequences  of  fifths  or  fourths 
were  only  possible  in  so  far  as  the  melos  lay  within  the  lower  tetrachord 
of  the  octave;  for  in  their  music  the  only  distinctive  melody  lay  below 
the  accompaniment.  'After  the  highest  available  fourth  had  been  reached, 
however,  the  accompaniment  remained  stationary  in  the  highest  tone 
of  the  instrument,  while  the  melody  wandered  at  will  even  into  unison 
with  it.  When  the  melody  again  descended  out  of  this  region,  it  drew 
the  accompaniment  with  it  in  fifths  or  fourths,  only  the  final  interval 
being  always  the  octave  (16, 232  fi.).  According  to  Aristotle  music 
involving  different  intervals  ('symphonic'  style)  was  less  pleasant  than 
the  antiphonic. 

Greek  music  thus  seems  to  have  been  essentially  monomelodic. 
Vocal  melody  was  evidently  absolutely  single  (cf.  16,  157).  And  although 
in  the  instrumental  style  a  further  approach  was  made  to  polyphony, 
especially  when  different  intervals  became  obligatory  in  the  upper 
tones,  yet  it  was  clear  to  the  Greek  ear  that  the  melos  still  lay  unobscured 
below  the  accompaniment.  The  latter  did  not  itself  form  a  voice  (16,  234). 

This  state  of  the  art  forms  a  most  interesting  parallel  to  the  earliest 
forms  of  Western  music  known  as  organum.  In  its  strict  form  this 
consisted  simply  of  series  of  fifths  or  of  fourths,  or  of  these  primary 
voices  doubled  at  the  octave,  the  upper  an  octave  below  and  the  lower 
an  octave  above.  The  very  difficulty  that  probably  prevented  the 
Greeks  from  magadising  in  fifths  or  fourths,  namely  the  occurrence 
of  a  tritone  instead  of  a  fifth  or  a  fourth  once  in  the  complete  scale, 
may  have  been  responsible  for  the  development  of  a  'free'  organum, 
in  which  the '  vox  principalis '  moved  from  unison  with  the '  vox  organalis ' 
up  to  the  fourth  while  the  latter  remained  stationary,  and  the  like 
(81,  51  £f.).  The  variant  thus  attained  was  then  preferred  for  its  own 
sake  and  developed  to  greater  freedom.  And  after  a  time  the  only 
other  possible  relation  of  voices — that  of  contrary  motion — seems  to 
have  appeared  quite  suddenly  (81,  71  ff.)  : 

The  earliest  known  expositions  of  the  new  organum  are  contained  in  the  Musica 
of  Johannes  Cotto,  written  about  the  year  1100....  The  organum,  we  find,  is  now 
constructed  entirely  of  consonances,  and  the  arrangement  of  these  is  decided  chiefly 
by  the  various  kinds  of  progression  adopted  by  the  voices....  Although  the  similar 
[parallel]  motion  of  the  voices  is  by  no  means  forbidden,  a  contrary  progression 
is  on  the  whole  preferred  (81,  77).  (Hie  facillimus  ejus  usus  est,  si  motuum  varieta.s 
diligenter  consideretur:  ut  ubi  in  recta  modulatione  est  elevatio,  ibi  in  organica  fiat 
depositio  et  e  con  verso.     83,  vol.  150,  1429.) 


XI]  CONSECUTIVE  FIFTHS  83 

But  the  series  of  consecutive  consonances  of  the  same  kind  did  not 
go  beyond  two  or  three.  "Existing  compositions  prove  that  the  first 
actual  expansion  of  the  polyphonic  principle,  the  addition. of  a  third 
part  to  the  original  two,  dates  from  this  period,  and  that  the  fourth 
part  followed  soon  after"  (81,  85;  cf.  44,  79). 

There  cannot  be  the  slightest  doubt  that  music  in  three  or  four 
parts  in  which  contrary  motion  prevails  is  polymelodic  or  polyphonic. 
There  is  not  now  any  such  difficulty  in  following  the  various  voices 
as  Plato^  and  Aristotle  complained  of  in  the  Greek  music  of  mixed 
intervals  (16, 149 f.).  And  it  is  a  noteworthy  fact,  which  our  further 
analysis  ^vill  illuminate  in  a  far-reaching  way,  that  the  decline  of  the 
antiphonic  style  and  the  gradual  emergence  of  the  prohibition  of  succes- 
sive fifths  and  octaves,  etc.,  proceeded  in  close  relation  to  the  develop- 
ment of  distinctive  polyphony. 

Thus  it  appears  that  musical  art  can  proceed  only  a  little  way 
before  it  comes  to  a  distinct  apprehension  of  the  bad  effect  of  consecutive 
fifths  and  before  it  makes  their  prohibition  a  primary  principle  of  con- 
struction. We  are  not  by  any  means,  however,  compelled  to  suppose 
that  early  Greek  and  Western  musicians  took  perverse  pleasure  in 
ill-sounding  experiments  in  symphonious  singing.  An  isolated  perfect 
consonance  has  at  all  times  a  beautiful  aspect  that  reveals  itself  very 
readily  to  the  mind,  although  comparison  with  some  other  intervals — 
when  they  have  been  found  and  fully  appreciated — may  make  it  seem 
thin  and  poor.  But  to  the  natural  uncritical  ear  a  high  grade  consonance 
is  beautiful.  If  that  beauty  is  made  the  object  of  great  attention,  it 
is  possible  that  it  might  maintain  itself  for  some  time  in  forms  of  usage 
that  at  the  same  time  presented  latent  aspects  of  ugliness.  We  have 
only  to  suppose  that  the  latter  had  not  yet  caught  the  attention.  Besides, 
this  trend  of  attention  would  be  prevented  by  another  feature  of  con- 
sonances— the  unity  of  voice  or  tune  to  which  they  in  their  grade 
approximate.  The  voices  of  men  and  women  singing  the  same  melody 
will  fall  into  octaves  because  of  the  ease  and  unity  thus  established. 
The  voices  of  women  or  of  men,  if  they  differ  from  one  another  in  pitch 
considerably,  wovdd  for  the  same  reason  tend  to  fall  into  the  next 
greatest  consonance — the  fifth, — as  the  octave  would  not  lie  near  enough 
to  their  average  difference  to  attract  their  voices  to  itself;  or  if  it  did, 
one  or  both  voices  might  be  subjected  to  too  much  strain.    Untrained 

*■  Plato  perhaps  was  not  specially  gifted  musically  (cf.  66,  n).  But  it  does  not  require 
any  exceptional  musical  faculty  to  follow  two  simultaneous  melodies,  if  they  have  been 
properly  composed  and  performed. 

6—2 


U  CONSECUTIVE  FIFTHS  [ch. 

singers  have  been  heard  to  sing  in  fifths.  Stumpf  recorded  this  of  two 
maids  at  work  in  his  domestic  kitchen  (52,  239).  In  primitive  music 
also  sequences  of  fifths  have  been  variously  established  (ibid.). 

An  interesting  experiment  in  such  music  has  been  recorded  by 
Gevaert  (1895,  15,  423)  : 

Sequences  of  fifths,  produced  without  thirds  and  performed  slowly  by  very 
true  voices,  have  nothing  disagreeable  about  them.  Consecutive  fourths  produce 
at  first  a  bizarre  effect,  but  the  ear  soon  accustoms  itseK  to  that.  I  made  a  peraonal 
experiment  with  this  on  the  8th  of  July,  1871,  at  an  archaeological  gathering  arranged 
by  my  friend  Aug.  Wagener,  the  eminent  hellenist,  in  the  ruins  of  the  Abbey  of  St 
Bavon  at  Ghent.  On  this  occasion  I  had  a  choir  of  men  and  children  perform  several 
diaphonic  specimens  of  the  two  species  [strict  and  free  Organum];  the  impression 
made  on  the  audience,  about  a  hundred  persons,  was  profound.  Everyone  was 
unanimous  in  finding  in  this  threadbare  harmony  a  penetrating  atmosphere  of  very 
remote  antiquity.  It  is  true  that  the  place  lent  itself  admirably  to  an  evocation  of 
this  nature.  g 

If  we  can  thus  show  why  sequences  of  fifths  are  for  some  time  in 
the  earliest  stages  of  the  art  not  only  tolerable,  but  more  or  less  inevitable, 
we  must  endeavour  to  find  out  on  what  basis  the  unpleasant  effect 
rests  that  soon  appears  in  the  further  development  of  the  art.  This 
problem  has  long  been  the  object  of  inquiry  and  debate,  and  it  is  well 
that  we  should  consider  carefully  what  grounds  of  explanation  have 
already  been  advanced. 

The  following  are  the  chief  theories  of  the  prohibition  : 
1.  Habit  and  tradition.  This  theory  was  advocated  by  W,  Pole 
(50,  283  ff.;  17,  113  f.).  A  ready  reason  for  the  prohibition  of  consecutive 
octaves  is  found  in  the  fact  that  counterpoint  is  a  series  of  different 
melodies  going  together.  Using  sequences  of  octaves  in  counterpoint 
thus  means  professing  to  keep  melodies  different  throughout  and  yet 
not  doing  so.  But  the  rule  as  to  fifths  has  always  been  a  great  puzzle, 
he  says  : 

It  is  asserted  and  generally  believed  that  there  is  something  naturally  repugnant 
to  the  ear  in  such  successions....  But  still  it  is  undeniable  that  any  series  of  musical 
sounds  will  be  accompanied  naturally  by  consecutive  fifths  as  well  as  by  consecutive 
octaves;  and  with  this  example  in  nature  before  us,  it  certainly  seems  difiicult  to 
say  that  such  sequences  are  forbidden  by  natural  laws. 

We  are  bound  to  distrust  here  the  appeal  to  the  ear....  It  cannot  be  denied  that 
a  succession  of  perfect  fifths  in  counterpoint  soimds  very  objectionable  to  musicians. 
But  it  must  be  recollected  that  from  the  first  moment  any  musician  began  to  study 
composition,  he  was  taught  to  hold  consecutive  fifths  in  abhorrence;  and  it  is  to  be 
expected  that  the  result  of  this  must  be  to  make  him  believe  that  they  are  naturally 


XI]  CONSECUTIVE  FIFTHS  85 

objectionable.  If  there  is  really  any  physical  or  physiological  cause  for  the  antipathy, 
it  ought  to  be  capable  of  being  shotvn;  if  it  cannot  be  shown,  we  have  a  right  to 
presume  it  is  merely  the  eflEect  of  education  and  habit....  We  know  one  thing  by 
experience,  namely,  that  these  fifths  do  not  sound  offensive  to  those  who  happen 
to  be  ignorant  of  the  rule  against  them.  There  are  many  persons  who  have  learnt 
music  practically,  and  have  been  accustomed  to  it  all  their  lives,  but  who  have 
never  had  a  lesson  in  harmony  or  composition;  and  if  such  people  attempt  to  write 
music  in  parts,  they  will  use  consecutive  fifths  without  the  slightest  hesitation,  and 
not  see  anything  objectionable  in  them; — rather  a  strong  argument,  it  would  seem, 
that  the  objection  arises  chiefly  from  a  knowledge  of  the  rule. 

Even  so  notable  a  writer  as  F.  A.  Gevaert  has  given  support  to  a 
kindred  view  in  writing  :  "Influenced  by  the  school  rule  that  prohibits 
the  succession  of  several  perfect  consonances  of  the  same  species,  the 
musicologists  have  not  failed  to  declare  the  diaphonies  of  the  epoch 
of  Hucbald  and  Guido  as  intolerable  and  monstrous.  That  is  a  counter- 
pointist's  prejudice"  (15,  423);  and  :  "it  is  a  modern  prejudice  to  believe 
that  sequences  of  consonant  fifths  as  such  jar  on  the  ear"  (16,  158).  In 
support  of  this  he  tells  of  the  experiment  already  quoted  (p.  84,  above), 
but  he  does  not  add  any  further  justification  of  his  view. 

Pole's  theory  is,  of  course,  very  extreme  and  may  be  opposed  on 
every  count.  As  C.  Stephens  pointed  out,  the  harmonic  fifth  is  so 
prominent  in  certain  cases,  e.g.  on  stopped  organ  pipes,  which  give 
the  alternate  harmonics,  that  it  may  make  that  timbre  unsuitable 
with  music  that  would  tend  to  direct  attention  to  its  presence,  e.g. 
when  a  fugal  subject  is  being  given  out  in  the  lower  part  of  the  instrument 
(17,  115).   G.  A.  Macfarren  declared  that  the  sequence  of  fifths  is 

repugnant  to  us  at  the  present  time,  and  not  in  this  room  alone,  not  in  this  country, 
but  throughout  all  the  civilised  world  wherever  music  is  studied,  and  wherever  it 
has  resolved  itself  into  a  language  instead  of  the  barbarous  jargon  of  savages.  I 
cannot  suppose  that,  as  long  as  the  organs  of  hearing  have  been  the  same,  persons 
can  have  experienced  pleasure  many  hundreds  of  years  ago  in  progressions  which 
are  entirely  offensive  to  us  who  hear  them  now;  that  the  same  acoustical  properties, 
whatever  they  may  be,  which  make  them  offensive  in  the  nineteenth  century  could 
have  been  absent  in  the  tenth  century;  and  that  progressions  which  through  their 
as  yet  undiscovered  properties  are  cacophonous  to  us  can  have  been  acceptable 
to  the  persons  who  heard  them  (17,  in). 

Of  course  many  persons  may  have  "learnt  music  practically  and  may 
have  been  accustomed  to  it  all  their  lives"  and  yet  may  never  have 
attended  to  it  analytically  or  aesthetically  at  all.  Just  as  there  are  so 
many  who  are  hardly  even  aware  of  the  diurnal  changes  in  the  colour 
of  objects  in  spite  of  their  having  attended  to  these  objects  in  many 
critical  practical  ways  which  would  seem  to  a  colour  artist  to   make 


86-  CONSECUTIVE  FIFTHS  [ch. 

such  ignorance  impossible.  Think  of  the  crudities  such  persons  would 
produce  in  a  first  attempt  at  water-colouring ! 

But  the  weakest  point  of  Pole's  position  is  that,  while  demanding 
from  the  defenders  of  the  prohibition  an  exposition  of  the  physical 
or  physiological  cause  for  the  antipathy,  he  omits  himself  entirely 
even  to  suggest  the  need  for  a  cause  of  the  convention,  as  he  thinks 
it,  by  which  in  the  first  instance  sequences  of  fifths  came  not  only  to 
be  forbidden  but  also  to  be  so  heartily  disliked.  Such  a  convention  to 
dislike  requires  as  strong  a  cause  as  any  unauthorised  hatred,  more 
especially  as  opposition  to  the  convention  does  not  seem,  even  in  Pole 
himself,  to  have  led  to  a  change  of  taste  for  consecutive  fifths. 

2.  Excessive  sweetness.  According  to  W.  H.  Cummings  (17,  114) 
John  of  Dunstable,  an  Englishman,  forbade  consecutive  fifths  : 

not  because  they  are  so  objectionable,  but  because  they  are  so  sweet,  so  that  the 
ancients  could  be  really  cloyed  with  the  sweetness  of  the  fifth.  We  know  that  fully 
to  the  end  of  the  thirteenth  century  most  of  the  harmony  we  can  find  consists  of 
fifths  and  octaves.  They  found  it  so  sweet  that  they  thought  it  was  time  to  leave 
it  oflf.   John  of  Dunstable  is  really  the  first  who  wrote  against  the  use  of  them. 

Or  as  G.  A.  Macfarren  expressed  it:  "John  of  Dunstable  said  they 
were  too  beautiful,  too  much  beauty  could  not  be  permitted,  therefore, 
a  succession  of  these  dehghts  was  overpowering  to  the  human  senses" 

(17,  118). 

Against  this  theory  Sacchi  wrote  (57,  6)  : 

No  one  ever  denied,  nor  shall  I,  that  successive  sweetness  can  change  to 
displeasure.  We  can  therefore  weU  understand  that  a  continued  series  of  ten  or  fifteen 
fifths  ought  to  displease  and  disgust  us;  and  that  would  not  be  improbable.  But 
that  one  single  repetition  of  the  fifth,  merely  by  reason  of  its  great  charm,  must 
suddenly  displease  and  offend  us,  is  not  intelUgible. 

That  is  quite  true.  But  we  have  not  only  to  refute  the  theory,  but 
to  account  for  its  formulation  as  well.  And  it  is  not  difl&cult  to  see  the 
motive  of  it.  The  fifth,  as  was  noted  above,  is  the  second  best  consonance 
or  fusion,  and  as  such  has  a  special  beauty  and  sweetness,  not  necessarily 
under  all  circumstances  of  comparison,  but  at  least  under  some.  Of 
these  circumstances  probably  only  the  latter  group  will  control  the 
judgment  of  the  primitive  ear  and  of  any  mind  that  is  for  the  time 
being  more  or  less  imcritical  or  forgetful  of  the  specially  interesting 
intervals  and  chords  that  music  has  developed.  In  any  case  there  is 
no  doubt  that  the  primitive  ear  is  fixed  upon  the  beauty  of  the  great 
consonances,  and  naturally  endeavours  to  make  its  art  out  of  these 


XI]  CONSECUTIVE  FIFTHS  87 

elements.  No  doubt  too  for  a  time  its  attention  to  them  prevails  over 
any  other  features  their  sequences  may  create.  But,  of  course,  these 
features,  being  so  pronounced,  soon  force  themselves  upon  the  attention. 
What  is  more  natural,  then,  than  that  the  only  known  feature  of  these 
intervals — their  great  consonance — should  be  taken  as  the  ground  of 
explanation,  and  that  the  theory  should  be  :  in  sequences  of  fifths 
there  is  too  much  of  the  fifth's  consonance?  It  is  not  that  the  ears  of 
these  early  folks  were  undeveloped  or  different  from  ours,  and  that 
their  minds  were  crude  and  lame;  but  their  attention  had  been  set 
into  a  certain  direction  by  the  course  of  the  art  till  their  time;  and  their 
minds  naturally  followed  its  suggestions.  Looking  backward  is  not 
nearly  so  difl&cult  as  groping  forwards  and  accommodating  soul  and 
mind  to  new  developments  quickly.  We  must  judge  leniently  when 
we  think  how  long  all  our  theories  have  tried  to  nourish  their  energies 
on  the  very  poor  diet  of  the  harmonics. 

3.  Want  of  variety.  Zarhno  wrote  in  1571  (82,  Part  iii,  chap.  29, 
p.  216)  : 

The  most  ancient  composers  forbad  the  placing  after  one  another  of  two  perfect 
consonances  of  the  same  genus  and  species  bounded  in  their  extremes  by  one  and 
the  same  proportion,  while  the  modulations  moved  by  one  or  more  steps;  as  the 
placing  of  two  or  more  unisons,  or  two  or  more  octaves,  or  two  or  more  fifths,  and 
such  like; ...  for  they  well  knew  that  harmony  cannot  spring  but  from  things  mutually 
diverse,  discordant,  and  contrary,  and  not  from  such  as  in  every  way  agree. 

The  composer,  he  says,  must  imitate  the  beauty  of  nature  which  makes 
no  two  things  of  any  species  exactly  alike. 

The  explanation  given  by  Helmholtz  includes  this  one  beside  others  : 

The  accompaniment  of  a  lower  part  by  a  voice  singing  an  octave  higher,  merely 
strengthens  part  of  the  compound  tone  of  the  lower  voice,  and  hence  where  variety 
in  the  progression  of  parts  is  important,  does  not  essentially  differ  from  a  unison. 
Now  in  this  respect  the  nearest  to  an  octave  are  the  twelfth,  and  its  lower  octave 
the  fifth.  Hence,  then,  consecutive  twelfths  and  consecutive  fifths  partake  of  the 
same  imperfection  as  consecutive  octaves  (20,  S89). 

Only,  the  case  is  worse  because  the  accompaniment  cannot  be  carried 
out  consistently  without  changing  the  key.  (The  a  of  the  key  of  c 
is  familiarly  a  little  flatter  than  the  just  a  of  the  key  of  d:  thus  two 
just  fifths  c-g  and  d-a  would  mean  a  departure  from  the  scale  of  c  in 
the  a.)  "Hence  an  accompaniment  in  fifths  above,  when  it  occurs 
isolated  in  the  midst  of  a  polyphonic  piece,  is  not  only  open  to  the 
charge  of  monotony,  but  cannot  consistently  be  carried  out"  (20,  360). 


88  CONSECUTIVE  FIFTHS  [ch. 

Helmholtz  then  proceedis  to  explain  that : 
when  the  fifths  are  introduced  as  merely  mechanical  constituents  of  the  compound 
tone,  they  are  fully  justified.  So  in  mixture  stops  of  the  organ....  It  would  be  quite 
different  if  we  collected  independent  parts,  from  each  of  which  we  should  have  to 
expect  an  independent  melodic  progression  in  the  tones  of  the  scale.  Such  independent 
parts  cannot  possibly  move  with  the  precision  of  a  machine;  they  would  soon  betray 
their  independence  by  sUght  mistakes,  and  we  should  be  led  to  subject  them  to  the 
laws  of  the  scale,  which,  as  we  have  seen,  render  a  consistent  accompaniment  in 
fifths  impossible. 

For  the  same  reasons  the  second  inversion  of  the  major  common  chord 
"expresses  a  single  compound  tone  much  more  decidedly  than  'the 
first  inversion,'  which  is  often  allowed  to  be  continued  through  long 
passages,  when  of  course  the  nature  of  the  thirds  and  fourths  varies" 
(20, 360).  The  second  inversion  may  be  represented  as  the  third, 
fourth,  and  fifth  partials  of  a  compound  tone,  the  first  inversion  as 
'only'  the  fifth,  sixth,  and  eighth.  But,  as  F.  E.  Gladstone  pointed  out 
(17, 102),  this  argument  will  not  apply  to  minor  chords.  The  minor 
chords  have  always  been  a  thorn  in  the  flesh  of  the  harmonic  derivations 
of  music. 

The  theory,  then,  argues  that  close  approximation  to  the  constitu- 
tion of  a  tonal  blend  of  fundamentals  and  partials  makes  sequences 
of  fifths  and  fourths  admissible.  These  sequences  are  forbidden  only 
between  distinct  parts,  because  we  expect  independence  and  variety 
from  them,  not  monotony  (cf.  Gladstone,  17,  105).  The  theory  suggests, 
but  does  not  state  explicitly,  that  the  prohibition  of  consecutives  is 
the  stricter  the  nearer  the  interval  in  question  lies  to  the  fundamental 
component  of  a  blend.  Thus  the  fourth  is  prohibited,  "but  with  less 
strictness"  than  the  fifth.  "Even  thirds"  have  been  forbidden  as  an 
accompaniment  (cf.  ibid.).  The  theory  is  apparently  consistent  logically. 

What  is  hard  to  understand  is  how  the  relation  to  partials  creates 
such  unpleasantness  in  this  case  while  in  single  consonances  it  makes 
for  harmony  and  pleasantness.  Of  course  every  theory  must  appear 
in  dealing  with  this  problem  to  pull  from  the  storehouse  of  explanation 
contrary  results  for  what  seem  very  similar  objects.  That  is  a  mere 
restatement  of  the  fact  that  isolated  consonances  are  pleasant,  while 
sequences  are  often  ugly.  What  every  theory,  however,  must  avoid 
doing  is  using  the  same  unaltered  ground  as  an  explanation  of  contrary 
results.  And  that  Helmholtz  seems  to  do.  If  "hearing  again  a  part  of 
what  we  heard  before"  is  a  ground  of  consonance,  it  is  unlikely  that 
this  alone  would  produce  the  ugliness  of  sequences  of  fifths.  Monotony 
is  an  idea  hardly  adequate  to  the  effect  to  be  explained.    As  Sacchi 


XI]  CONSECUTIVE  FIFTHS  89 

might  have  said  :  we  could  understand  that  ten  or  fifteen  fifths  in 
sequence  would  have  been  boredom,  but  we  should  hardly  take  ofEence 
at  two.  Helmholtz  seems  to  have  felt  this  himself  somewhat;  for  he 
inclines  in  part  to  Pole's  view,  saying  : 

The  prohibition  of  consecutive  fifths  was  perhaps  historically  a  reaction  against 
the  first  imperfect  attempts  at  polyphonic  music,  which  were  confined  to  an  accom- 
paniment in  fourths  or  fifths,  and  then,  like  all  reactions  it  was  carried  too  far, 
in  a  barren  mechanical  period,  till  absolute  purity  from  consecutive  fifths  became 
one  of  the  principal  characteristics  of  good  musical  composition.  Modern  harmonists 
agree  in  allowing  that  other  beauties  in  the  progression  of  parts  are  not  to  be  rejected 
because  they  introduce  consecutive  fifths,  although  it  is  advisable  to  avoid  them 
when  there  is  no  need  to  make  such  a  sacrifice  (20,  seo). 

Here  he  has  not  reached  the  point  of  view  of  some  who  claim  that 
in  these  exceptional  cases  it  is  not  a  matter  of  admitting  the  ugliness 
for  tHe  sake  of  the  beauty,  but  of  outweighing  the  ugliness  so  that  it 
no  longer  appears  or  even  of  creating  positive  beauty  (cf.  35,  84). 

The  argument  from  want  of  variety  is  weak  in  so  far  as  it  has  to 
meet  its  own  objection  for  e\ery  interval;  including  sixths  which  by 
no  manner  of  means  can  be  claimed  as  forbidden  in  sequence  (cf. 
C.  Stephens,  17,  lis,  and  G.  A.  Macfarren,  17,  ii9,  who  refer  to  both 
thirds  and  sixths).   Sacchi  added  that 

it  is  true  that  on  the  false  principle  that  two  fifths  displease  for  lack  of  variety, 
some  [e.g.  Zarlino^]  have  drawn  the  false  conclusion  that  similarly  the  repetition 
of  thirds  and  of  sixths  ought  to  displease  when  they  are  of  the  same  species,  and 
have  therefore  forbidden  it;  but  their  prohibition  was  not  accepted  by  composers, 
who,  disciplined  by  experience,  carefully  avoid  repeating  fifths,  but  are  not  in  the 
least  concerned  about  the  repetition  of  thirds  or  sixths  of  the  same  species.  Vain 
is  therefore  likewise  the  reason  that  is  drawn  from  the  desire  for  variety,  which 
if  true  would  be  equally  so  in  all  the  consonances;  it  would  hold  rather  more  in 
the  imperfect  than  in  the  perfect;  because  after  all  it  ought  to  be  more  tolerable 
to  the  ear  to  linger  on  the  sweeter  consonance  than  on  the  less  sweet  (57,  ^  t). 

The  validity  of  the  argument,  therefore,  vanishes  entirely  in  so 
far  as  it  is  mere  variety.  Variety  is  certainly  desirable,  but  it  would 
be  as  pedantic  to  prescribe  it  at  every  instant  as  to  forbid  consecutive 
fifths  when  they  sound  well. 

If  by  variety  we  mean  specially  the  monotony  of  compound  tones, 

*  Non  si  debbe  anco  porre  due  6  piu  imperfette  consonanzo  insieme  ascendenti  6  dis- 
cendcnti  1'  una  dope  1'  altra  senz'  alcun  mezo;  come  sono  due  Terze  maggiori,  due  minori, 
due  Seste  maggiori  anco  e  due  minori.  Conciosiache  non  solo  si  fi  contra  quello  c'ho 
dotto  delle  Perfctte;  ma  il  loro  procedero  si  fi  udire  alquanto  aspro;  per  non  haver  nella 
lor  modulatione  da  parte  alcuna  1'  intervallo  del  Semitonio  maggiore,  nel  quale  consisto 
tutto  '1  buono  nclla  Musica,  e  senza  lui  ogni  Modulatione  ed  ogni  Harmonia  6  dura, 
aspra,  c  quasi  inconsonante  [82,  ai7). 


90  CONSECUTIVE  FIFTHS  [ch. 

surely  the  condensation  of  the  partial  components  into  the  range  of  an 
octave  would  make  a  sufficient  variation  from  the  compound  tone. 
And  the  inevitable  departure  from  the  justness  of  the  fifths  when  the 
music  remains  consistently  in  one  key  would  help  to  assuage  the 
monotony.  Consecutive  fifths  would  then  partly  cease  to  be  fifths, 
and  should  in  so  far  be  tolerable.  Besides,  if  two  consecutive  fifths  of 
different  pitch  involve  too  little  variety  and  are  therefore  forbidden, 
is  there  not  still  less  variety  in  the  repetition  of  one  and  the  same  fifth? 
And  yet  that  sequence  is  unobjectionable. 

Thus  it  appears  that  the  argument  is  insufficient  to  explain  the 
effect  produced  by  two  consecutive  fifths.  Some  much  more  decided 
difference  must  be  the  source  of  the  ugliness  in  question. 

We  shall  see  as  we  proceed  that,  if  the  word  'independence,'  which 
has  often  been  used  in  this  connexion  instead  of  the  word  'variety,' 
were  properly  emphasised  and  defined,  much  could  be  said  in  favour 
of  the  theory.  The  reader  may  therefore  in  the  end  feel  that  the  variety 
theory  has  much  in  its  favour.  No  doubt  those  who  advocated  it  felt 
this  underlying  justification,  but  in  their  expressions  they  refer  only 
to  independence  in  the  sense  of  variety  or  difference  of  voices.  They 
did  not  mean  by  independence  the  independence,  as  distinct  from  the 
mere  variety,  of  the  voices. 

4.  Ambiguity  of  hey  or  tonality.  One  of  the  theories  discussed  by 
Sacchi  falls  more  or  less  under  this  head,  although  it  is  not  quite  the 
same  as  the  form  most  famiHar  at  the  present  time.  The  prohibition 
of  consecutive  fifths  in  this  case  was  held  to  be  due  to  the  great  difference 
between  the  scales  that  arise  from  the  bases  of  the  successive  chords. 
Thus  the  scale  of  d  differs  from  that  of  c  in  two  notes  (the  two  sharps). 

But,  said  Sacchi,  why  not  imagine  in  the  scale  of  d  instead  of  a  major, 
a  minor,  third  (as  in  the  melodic  minor  scale).  Then  there  would  be 
a  difference  of  only  one  note,  i.e.  the  least  possible  difference,  between 
the  scales.  And,  after  all,  the  difference  alleged  is  not  really  heard, 
it  is  only  imagined,  or  at  least  conceived.  The  argument  surely  attributes 
too  great  force  to  the  imagination.  Besides,  the  four  tones  heard 
{c-g  and  d-u)  actually  belong  to  one  and  the  same  scale  and  have 
optimal  consonance  with  one  another.  How  then  could  they  be  turned 
into  an  offence  by  mere  imagination  or  rather  by  the  mere  possibility 
of  imagining  two  other  notes?  "This  will  certainly  in  no  wise  happen  : 
for  things  imagined,  and  that  can  be,  never  prevail  over  such  as  are 
or  are  felt"  (57,  8ff.). 


XI]  CONSECUTIVE  FIFTHS  91 

Although  the  theory  in  question  would  hardly  find  a  champion 
nowadays,  the  argument  against  the  force  of  imagination  is  noteworthy. 
Imagination  or  the  inclination  of  interpretation  is  often  very  important 
in  music,  but  it  is  well  to  be  reminded  that  such  a  fluctuating  and 
divertible  force  is  not  likely  to  be  the  cause  of  a  very  constant  and 
highly  undivertible  phenomenon,  such  as  the  unpleasantness  of  con- 
secutive fifths. 

The  theory  of  the  prohibition  advanced  by  Sacchi  himself  is  the  am- 
biguity of  tonality  created  by  the  sequence  of  fifths,  e.g.  c~g  and  d-a. 

I  have  no  sufficient  reason  to  refer  the  second  note  {d)  to  the  first  (c)  or  conversely 
the  first  to  the  second  (57,  76).  I  might  be  told  that  I  ought  to  refer  the  second 
nl  to  the  first,  because  the  sound  of  the  first  has  already  taken  possession  of  my  ear. 
But  with  equal  reason  I  might  be  told  that  the  first  ut  ought  to  refer  to  the  second, 
because  all  other  things  being  equal,  the  present  sensation,  being  more  lively,  is 
to  be  given  precedence  of  the  past.  The  suspension  is  therefore  imreheved;  and  I 
am  lost  in  ambiguity,  indetermination  and  suspense  between  two  different  notes, 
each  of  which,  without  any  difference,  can  be  considered  as  primary  base  {ibid.  f.). 

Those  who  follow  this  type  of  theory  do  not  usually  admit  that  all 
other  things  are  so  perfectly  equal  as  Saccbi  said.  And  if  they  are,  is 
there  not  an  obvious  means  of  overcoming  the  ambiguity — by  indicating 
the  key  with  aU  possible  precision  before  the  consecutive  fifths  are 
introduced?  Then  the  ground  of  objection  to  them  alleged  by  Sacchi 
would  be  removed.  If  imagination  is  an  insufficient  force  to  explain 
the  offensiveness  in  question,  we  must  surely  conclude  that  any  such 
ambiguity  is  only  very  slightly  more  potent.  An  easy  way  out  of  any 
such  difficulty  is  in  constant  evidence  in  every  exactly  measured  sequence 
of  objectively  equal  intervals  of  time.  The  attention  may  elect  to 
hear  this  sequence  as  one  or  other  of  various  rhythms :  "^ ',  ^ ',  "^ ' . . .  or 
'  ^/  ^/  ^  ...  or  ^  ^^/  ^^/  ^^  ...  and  so  on.  But  in  spite  of  the  great 
number  of  possibilities,  each  of  which  can  be  realised  at  inclination,  the 
ear  never  remains  tortured  by  ambiguity.  It  adopts  at  once  a  definite 
rhythm,  possibly  the  easiest  under  the  circumstances.  When  it  has 
had  enough  of  that  one,  it  'fluctuates'  into  another,  it  may  be.  If  the 
answer  be  that  such  an  involuntary  solution  is  impossible  in  dealing 
with  keys  because  they  are  different  from  mere  rhythms,  then  we  must 
reply  that  if  the  apprehension  of  tonality  is  at  all  difficult  and  not 
so  inevitable  as  is  rhythm,  then  the  mind  should  receive  the  sequent 
chords  without  any  'thought'  of  tonality  at  all, — as  an  unmusical  mind 
certainly  would. 

And  Sacchi  himself  excludes  the  last  possible  reply  at  this  point 
by  his  words : 


92  CONSECUTIVE  FIFTHS  [ch. 

In  fact  the  two  successive  fifths  not  only  offend  the  ear  of  the  erudite  in  music, 
but  of  those  even  who  have  no  practice  in  it,  so  long  as  they  happen  to  have  been 
born  with  a  good  and  subtle  ear,  and  pay  attention  to  what  they  hear.  For  where 
one  does  not  attend,  which  of  the  dissonances  will  not  pass  unobserved  and  without 
offending?  (57,  le). 

The  central  attitude  to  chords,  whereby  the  lowest  component  is 
the  most  sonorous,  most  before  the  attention,  etc.,  may  well  be  inevitable 
even  to  those  unpractised  in  the  ways  of  music;  but  no  one  could  well 
suggest  that  the  apprehension  of  tonality  is  such  an  inevitable  and 
'natural'  attitude,  requiring  no  practice  and  experience.  There  is  some 
evidence  that  the  Greeks  related  the  pitches  of  all  their  notes  to  one 
particular  note — the  mese, — at  least  for  the  purposes  of  tuning,  if  not 
with  some  feeling  for  'tonality.'  In  the  latter  case  the  functions  of  their 
tonic  must  at  least  have  been  very  different  from  ours.  Before  a  certain 
period  of  modern  music  the  sense  of  tonality  was  much  weaker  than 
it  is  now.    In  various  cases  it  may  have  hardly  been  present  at  all. 

Finally,  ambiguity  of  tonality  is  by  no  means  uncommon  in  more 
modern  music  at  least.  Transposition  from  one  key  to  another  is 
frequently  affected  by  means  of  one  or  more  chords  that  are  common 
to  both  keys.  In  a  familiar  piece  of  music,  then,  one  may  not  only  '  hear ' 
the  key  that  is  to  be  left  but  one  may  be  able  to  anticipate  that  to  come, 
so  that  the  transitional  chords  may  be  in  a  real  sense  ambiguous.  And 
yet  no  such  horrid  effect  is  thereby  produced  as  is  characteristic  of 
consecutive  fifths.  Considerable  dispute  is  often  possible  as  to  which 
key  a  short  passage  of  a  musical  piece  really  displays;  but  no  specially 
inartistic  effect  appertains  to  such  passages.  Mere  ambiguity  of  key 
without  any  other  difference  is  therefore  useless  as  an  explanation  of 
the  prohibition  of  two  sequent  fifths. 

Perhaps  no  explanation  is  more  frequently  offered  for  the  disagreeable  effect 
of  consecutive  fifths  than  that  suggested  by  Cherubini:  viz.  that  two  parts  moving 
progressively  by  fifths  are  moving  in  two  different  scales^^.  The  reason  is  obviously 
insufficient;  but  it  has  more  force  than  some  critics  are  willing  to  admit,  when,  for 
instance,  three  triads  in  succession  are  based  upon  the  notes  C,  D,  E,  the  first  having 

^  For  a  very  primitive  form  of  this  theory — in  the  writer  of  the  Commentary  called 
Scholia  Enchiriadis,  see  81,  58  f.:  "We  learn  that  it  is  the  impropriety  of  this  combination 
of  two  different  modes  or  species  of  the  scale,  throughout  the  whole  of  a  composition, 
which  in  his  view  gives  rise  to  the  necessity  for  a  free  treatment  [of  the  organum]." 
(Quare  in  Diatessaron  symphonia  vox  organalis  sic  absolute  convenire  cum  voce  princi- 
pali  non  potest,  sicut  in  symphoniis  aliis  ?  Quoniam  per  quartanas  regiones  non  iidem 
tropi  reperiuntur,  diversorumque  troporum  modi  per  totum  ire  simul  ire  nequeunt.  83, 
vol.  132,  1003;  cf.  p.  972.) 


XI]  CONSECUTIVE  FIFTHS  93 

a  major  third,  and  the  other  two  having  minor  thirds,  but  each  with  a  perfect  fifth, 
it  seems  clear  that  the  parts  do  not  all  progress  in  one  scale  throughout.  The  upper 
parts  cannot  really  be  in  the  scale  of  C,  because,  as  we  know,  neither  a  true  fifth 
nor  a  minor  third  can,  strictly  speaking,  be  based  upon  the  second  degree  of  an 
accurately  tuned  major  scale.  But  it  may  be  said  that  the  instrument  upon  which 
these  chords  have  been  played,  is  tuned, upon  the  system  known  as  'equal  tempera- 
ment.' No  doubt  it  is.  Nevertheless  I  contend  that,  as  we  tolerate  its  sharp  major 
thirds  and  flat  fifths,  knowing  and  feeling  them  to  be  substitutes  for  the  true  thirds 
and  fifths  of  the  genuine  scale,  so  we  are  accustomed  to  accept  other  divisions  of 
the  scale,  not  for  what  they  actually  are,  but  for  what  they  represent....  To  me, 
therefore,  it  does  not  seem  urureasonable  to  argue  that  even  with  the  pianoforte 
we  recc^nise  the  equivocal  nature  of  such  a  progression  as  that  contained  in  my 
first  example  [triads  in  C,  D,  ir\.  But  even  after  this  has  been  granted,  the  argument 
that  consecutive  fifths  cause  two  parts  to  move  in  different  scales  cannot  be  carried 
much  further.  The  triads  on  the  third,  fourth,  fifth,  and  sixth  degrees  are  all 
perfectly  in  tune  in  the  scale  of  just  intonation.  Some  further  reason,  therefore, 
must  be  sought  for  the  unquestionably  ugly  result  produced  when  these  chords 
are  taken  in  regular  rotation,  either  ascending  or  descending  (17,  loof,;  cf.  G.  A. 
Macfarren  (17,  ii9f.;  35,  10). 

The  argument  thus  properly  refuted  by  F.  E.  Gladstone  contains 
at  the  best  something  very  like  a  dilemma  which  renders  it  obscure 
and  confusing.  When  we  hear  fifths  true  to  the  scale  of  c  major  on 
c,  d  and  e,  our  musical  habit  may  make  us  do  either  of  two  things  : 
either  we  are  governed  by  our  habit  of  the  major  scale  on  c,  and  then 
the  intervals  are  heard  as  played  and  the  second  is  not  a  true,  but  only 
an  approximate,  fifth  and  is  heard  as  such;  or  we  are  drawn  away 
rather  by  the  habit  of  the  perfect  fifth,  when  the  intervals  will  not  be 
heard  as  they  are  played,  but  only  as  they  suggest — in  perfect  form — and 
the  sequence  will  lie  in  no  one  scale.  If  the  former  alternative  is  valid, 
the  prohibition  of  consecutive  fifths  must  hold  for  all  fifths,  whether 
perfect  or  approximate,  if  the  ugly  effect  persists  in  spite  of  the  recognised 
approximation  (and  that  it  does  persist  could  hardly  be  denied).  The 
alternative  would  only  prohibit  such  fifths  as  lead  to  a  distortion  of 
the  intended  scale.  A  third  alternative  might  claim  that  we  hear  all 
the  tones  and  intervals  as  played,  but  that  we  take  the  approximate 
fifths  as  representing  true  ones  and  are  disturbed  by  the  distortion 
of  scale  thereby  implied.  This  would  surely  be  a  needless  procedure 
in  view  of  the  fact  that  even  if  we  take  approximate  thirds  as  representing 
just  thirds  (minor),  we  do  not  thereby  feel  any  distortion  of  scale. 

Of  these  three  alternatives  the  first  seems  not  only  easier  and  more 
natural  but  also  most  in  accordance  with  the  whole  system  of  musical 
synthesis  and  apprehension.   It  is  the  only  one  which  makes  the  system 


94  CONSECUTIVE  FIFTHS  [oh. 

of  equal  temperament  musically  tolerable  for  permanent  use  or  at 
least  as — for  any  individual — the  only  known  system.  For  the  other 
alternatives  presuppose  that  the  pianoforte  is  only  tolerable  in  virtue 
of  the  just  scales  it  suggests,  which  must  have  been  otherwise  in- 
eradicably  planted  in  the  minds  of  all  those  who  use  that  instrument. 
That  is  probably  true  of  those  whose  experience  has  made  them  most 
familiar  with  instruments  that  play  just  intervals.  But  for  the  great 
majority  the  intervals  their  minds  apprehend  when  the  piano  is  played 
are  those  actually  played,  the  scale  known  is  the  pianoforte  scale,  and 
the  rules  and  prohibitions  of  musical  structure  are  valid  for  all  these 
approximations. 

It  is  important  to  notice  that  these  approximations  are  primarily 
matters  of  consonance  and  its  grades.  An  equal  fifth  partakes  as  a 
consonance  of  the  grade  of  fusion  found  optimally  in  the  just  fifth. 
The  same  holds  still  more  for  the  dissonances.  And  as  intervals  the 
latter  can  be  learnt  in  their  approximate  form;  the  just  form  of  the 
dissonant  intervals  has  no  such  special  nature  as  a  fusion  as  would 
enable  it  to  draw  the  ear  towards  itself.  It  is  a  familiar  fact  that  the 
high  grade  consonances  have  this  power.  It  is  difficult  even  to  rtrike 
the  dissonances  that  lie  near  these  consonances  because  the  voice  tends 
to  slip  into  the  easier  and  more  familiar  consonance. 

A  special  objection  to  the  key  theory  lies  in  the  fact  that  consecutive 
fifths  are  only  objectionable  when  they  lie  between  the  same  voices 
of  the  music.  That  is  a  primary  condition  of  the  phenomenon,  but  it 
is  by  no  means  a  condition  of  sameness  of  key.  In  music  of  any  definite 
tonality  all  the  voices  of  any  moment  are  held,  and  are  intended,  to 
be  in  the  same  key,  whatever  that  may  be.  There  is  no  recognised 
complication  of  tonality  in  which  a  group  of  keys  are  considered  to  be 
maintained  concurrently,  one  in  each  separate  voice,  or  pair  of  voices. 
Hence  the  key  theory  loses  its  ground  entirely. 

It  is  true  that  in  recent  compositions  parts  or  groups  of  parts  have 
sometimes  been  made  to  move  concurrently  in  different  keys  (cf.  23, 
138  f.).  But  that  is  quite  a  different  matter.  Each  group  of  parts  is 
still  subject  to  the  fundamental  rules  of  harmony,  although  as  between 
the  separate  groups  the  claims  of  these  are  largely  ignored. 

5.  Want  of  relationship.  The  theory  advanced  by  Gladstone  is 
"  that  consecutive  fifths  are  generally  more  or  less  offensive  in  proportion 
to  the  want  of  relationship,  or  otherwise,  existing  between  the  chords 
which  produce  them."    He  cited  Kollmann  as  probably  the  first  writer 


XI]  CONSECUTIVE  FIFTHS  95 

who  propounded  this  idea^.  Unfortunately  his  case  was  spoilt  at  the 
very  outset  by  his  admission  that  there  is  as  little  relationship  between 
the  inversions  of  chords  on  successive  notes  of  the  scale  as  between 
their  original  positions;  and  of  the  inversions  a  succession  of  six- three 
chords  is  quite  admissible.  The  objection  attaching  to  the  six-fours 
does  not  alter  this  fact  in  the  least,  nor  its  destruction  of  the  theory 
of  relationship.  We  may,  therefore,  expect  the  frequency  with  which 
consecutive  fifths  are  admitted  between  tonic  and  dominant  chords 
and  vice  versa,  or  between  tonic  and  subdominant  chords  and  vice 
versa,  to  have  some  other  cause.  We  cannot  interpret  this  frequency 
as  due  solely  to  the  high  degree  of  relationship  between  the  chords  in 
the  sense  of  relationship  implied  when  we  say  that  the  triads  on  C  and 
D  are  totally  unrelated.  Gladstone  admitted  besides,  that  he  had  "  not 
met  with  any  specimens  of  consecutive  fifths  in  which  the  roots  of  the 
chords  rise  a  third  (except  where  a  sudden  change  of  key  occurs)" 
(17,  KM),  and  evidently  had  found  only  two  cases  of  a  fall  of  a  third — 
tonic  to  submediant. 

Gladstone  proceeded  to  note  that  the  objection  might  be  raised 
that  this  argument  ought  also  to  apply  to  fourths,  thirds,  and  sixths; 
and  recalled  in  reply  that  the  movement  of  fourths  is  placed  under 
various  restrictions  by  the  laws  of  counterpoint,  and  that  even  two 
major  thirds  in  succession  are  still  forbidden  in  the  strictest  style  of 
two-part  writing  (17,  105),  But  however  interesting  this  extension  of 
the  basis  of  argument  may  be,  it  is  perfectly  obvious  that  the  unexplained 
relaxation  of  the  prohibition  in  these  cases  only  makes  the  theory  of 
relationship  the  more  impossible. 

6.  The  nature  of  the  interval  itself.  The  study  of  the  connexion 
between  consecutive  fifths  (and  octaves)  and  the  harmonic  relationship 
of  chord?  has  been  renewed  by  Shinn  (58,  2C5ff.;  59).  But  Shinn 
does  not  attempt  to  explain  the  prohibition  of  successive  fifths  and 
octaves  by  the  lack  of  harmonic  relationship  between  the  chords  in 
which  they  stand,  but  by  the  intrinsic  character  of  the  intervals  of  the 

'  It  was  also  tho  basis  of  Pearsall's  (1795-1856)  explanation  who  wrote  that 
"consecutive  great  thirds  and  perfect  fifths  are  evidences  that  some  harmony  has  been 
sprung  over  which  ought  to  have  been  introduced  by  its  characteristic  note,  as  forming 
the  natural  link  of  relationship  between  these  intervals."  When  they  are  not  evidences 
of  such  a  spring,  "they  carry  with  them  an  awkwardness  of  progression  which  ought 
to  be  avoided."  They  display  "a  want  of  freedom  and  a  clumsiness,  unacceptable  to  any 
musical  ear"  (49,  as).  But  not  even  strong  disapproval  forms  a  logical  complement  to 
an  incomplete  theory. 


96  CONSECUTIVE  FIFTHS  [ch. 

octave  and  the  fifth  themselves.  For  the  octave  Shinn  adopts  the 
explanation  that  may  be  termed  traditional.  The  bad  effect  of  a  '  hidden ' 
octave  and  a  fortiori  of  consecutive  octaves  is  "the  weakness  which 
is  produced  by  the  correspondence  in  sound  of  the  two  outside  parts 
(approached  in  this  manner) "  (58,  268).  This  statement  revives  the 
explanation  by  want  of  variety  discussed  above.  The  bad  effect  of  a 
hidden  fifth,  and  a  fortiori  of  consecutive  fifths  is  due  "to  the  bareness 
of  the  interval"  (58,  268,  280). 

This  expression  may  be  taken  either  as  a  variant  upon  the 
"weakness"  of  the  octave  or  as  a  new  kind  of  reason  that  has  not  been 
advanced  by  any  other  theorist,  as  far  as  I  am  aware.  It  makes  perhaps 
some  approach  towards  the  notion  of  consonance  as  approximate  unity. 
But  obviously  it  is  now  an  aspect  of  the  interval  (or  fusion)  of  the 
fifth  that  displeases,  whereas  in  the  octave  it  is  the  correspondence  in 
sound  of  the  two  tones  that  make  up  the  interval — the  so-called  identity 
or  similarity  of  octave-tones.  Such  a  difference  of  causes  could  hardly 
be  acceptable. 

It  is  not  easy  to  give  a  just  systematic  place  to  Shinn's  exposition. 
In  some  ways  he  suggests  the  first  theory  of  this  chapter.  He  says, 
for  example  (58,  263)  : 

C!ombinations  and  progressions  which  were  formerly  regarded  as  painfully  crude, 
harsh  and  ugly,  have,  by  familiarity,  lost  these  characteristics,  and  become  both 
piquant  and  pleasant;  while  others,  which  had  hitherto  produced  pleasure,  now 
seem  commonplace  in  comparison  with  the  poignancy  of  less  familiar  but  more 
forcible  ones.  In  connexion  with  this  matter,  the  important  point  to  be  recognised 
is,  that  no  change  has  taken  place  in  the  progressions  themselves,  but  it  is  the  ear 
of  the  listener  which  has  changed,  owing  to  the  influence  of  a  change  in  his  musical 
environment  (68,  aes). 

Elsewhere  he  speaks  of  the  "so-called  objectionable  effect  of  con- 
secutive fifths"  (p.  284). 

On  the  other  hand  he  points  out  that  this  bad  effect  (whether  so- 
called  or  not)  is  "almost  invariably  neutralised  by  the  harmonic 
relationship  which  exists  between  the  chords  forming  such  fifths" 
{ibid.).  The  bareness  of  the  fifth,  we  are  to  understand,  is  somehow 
annulled  or  enriched.  But  Shinn  neither  explains  why  the  fifth  is  a 
bare  interval — I  suppose  it  just  sounds  so — nor  does  he  show  how  chord 
relationship  removes  this  bare  character  from  the  interval.  In  fact  he 
is  not  always  quite  faithful  to  the  explanation  by  harmonic  relationship 
and  abandons  it  in  part  in  favour  of  "the  effect  of  musical  strength 
which  is  characteristic  of  (such)  progressions"  of  the  two  voices  con- 


XI]  CONSECUTIVE  FIFTHS  97 

cerned  by  a  fourth  and  a  fifth  respecti\ely  (58,  277).  It  is  not  entirely 
a  matter  of  "harmonic  relationship  (or  root  progression),  but  partly 
also  of  the  movement  of  the  outside  parts  by  almost  equal  intervals." 
The  latter  acts  even  without  any  special  harmonic  relation. 

Nevertheless  Shinn's  discussion  is,  as  we  shall  see  later,  probably 
the  most  'philosophical'  one  that  has  so  far  been  given. 

7.  Want  of  balance.  A  suggestion  made  by  G.  A.  Macfarren  is 
worthy  of  mention,  although  it  has  not  been  worked  out  into  a  definite 
theory  as  far  as  I  am  aware.  "When  a  passage  of  harmony  in  any 
number  of  parts  has  two  notes  made  so  very  much  more  prominent 
than  the  rest,  as  is  the  case  in  the  duplication  of  those  two  at  the  expense 
of  the  others,  the  other  portion  of  the  harmony  is  enfeebled,  and  the 
balance  is  destroyed"  (17,  us).  This  is  exemplified  in  the  case  of 
successive  octaves,  when  two  notes  mutually  reinforce  each  other  and 
so  become  particularly  prominent  over  against  the  rest  of  the  score. 
The  same  does  not  hold  for  the  doubhng  of  a  voice,  even  although  the 
relation  is  now  merely  two  to  three  instead  of  two  to  two;  for  here  the 
doubled  part  is  meant  to  be  specially  prominent. 

In  this  restricted  form  the  theory  of  balance  is  not  very  significant. 
For  the  balance  in  question  is  chiefly  a  balance  of  mere  strength.  An 
overbalance  of  strength  can  easily  be  produced  in  music  and  often 
occurs  not  only  by  mere  accident  and  through  the  imperfect  technique 
of  performers,  but  through  want  of  finish  on  the  part  of  the  composer. 
On  none  of  these  occasions  could  it  well  be  said  to  be  so  strikingly 
unpleasant  as  to  justify  the  view  that  consecutive  octaves  (and  fifths) 
are  forbidden  so  strictly  because  of  the  disproportion  of  strength  they 
produce. 


CHAPTER  XII 

THE  SYSTEM  OF  FACTS  REGARDING  CONSECUTIVES 

The  preceding  chapter  contained  a  review  of  the  explanations  that 
have  been  attempted  for  the  prohibition  of  consecutive  fifths.  We  have 
noticed  how  the  various  theories  try  to  set  the  prohibition  into  relation 
to  facts  of  a  similar  kind  so  as  to  obtain  some  indication  of  systematic 
coherence  in  the  explanation.  None  of  the  systems  of  facts  thus  suggested 
is  very  satisfactory  or  convincing ;  and  none  of  the  theories  can  possibly 
be  held  to  be  successful.  It  is  extremely  doubtful  whether  anyone  who 
has  thus  far  reflected  on  the  problem  of  consecutive  fifths  has  felt  that 
more  than  interesting  suggestions  towards  an  explanation  have  been 
reached. 

"Often  and  often  have  I  thought,"  said  G.  A.  Macfarren,  "it  would 
require  the  entire  knowledge  of  a  physicist  to  be  able  to  probe  this 
subject  to  its  foundation"  (17, 119).  But  the  day  when  physical  science 
may  be  expected  to  solve  such  a  problem  is  now  definitely  past.  Even 
Helmholtz,  whose  basis  of  explanation  might  well  seem  to  many  to  be 
physical,  was  perfectly  well  aware  that  the  ground  of  explanation  of 
all  musical  phenomena  must  lie  within  the  phenomenal  stuff  of  sound 
itself;  it  dare  not  be  merely  physical  (20,  231  f.,  368).  No  doubt  the 
prominence,  or  at  least  the  great  propinquity,  of  the  physical  throughout 
his  exposition  prevented  many  of  his  followers  from  giving  sufficient 
heed  to  the  psychical  or,  if  you  like,  to  the  phenomenal  aspect  of  the 
problems  of  music.  Besides  the  difficulty  and  apparent  obscurity  of 
the  psychical  itself  made  them  only  too  eager  to  seize  upon  any  plausible 
excuse  for  evading  the  study  of  its  elementary  aspects.  Such  an  excuse 
was  not  only  given,  but  even  emphasised  by  Helmholtz. 

The  system  of  scales,  modes,  and  harmonic  tissues  does  not  rest  solely  upon 
inalterable  natural  laws,  but  is  also,  at  least  partly,  the  result  of  esthetical  principles, 
which  have  already  changed,  and  will  still  further  change,  with  the  progressive 
development  of  humanity  (20,  235). 

This  proposition,  he  said,  was  "not  even  now  sufficiently  present 
to  the  minds  of  our  theorists  and  historians."  But  ever  since  then,  at 
least,  it  has  been  decidedly  obstructive  in  its  effect  upon  their  minds. 
It  was  a  most  unfortunate  dictum.  For  the  opposition  implied  in  it 
between  natural  laws  and  aesthetical  principles  strongly  suggested  that 


CH.  xn]  REGARDING  CONSECUTIVES  99 

the  latter  are  merely  arbitrary  conventions,  as  is  more  or  less,  for  example, 
the  fashion  of  clothes  in  any  year.  After  quoting  this  dictum  Prout 
wrote  : 

While,  therefore,  the  author  [himself]  follows  Day  and  Ouseley  in  taking  the 
harmonic  series  as  the  basis  of  his  calculations,  he  claims  the  right  to  make  his  own 
selection,  on  aesthetic  grounds,  from  these  harmonics,  and  to  use  only  such  of  them 
as  appear  needful  to  explain  the  practice  of  the  great  masters  (52,  1st  ed.,  1889  iv). 

And  many  others  besides  Prout  could  be  quoted  to  the  same  ejffect. 

But  an  aesthetical  principle  is  not  the  sort  of  thing  that  men  for 
centuries  in  vain  seek  to  explain.  So  hidden  a  cause  is  rather  an  aesthetical 
law, — which  is  just  as  much  law  as  is  any  physical  uniformity.  And  it 
can  no  more  be  laid  aside  in  this  arbitrary  way  than  an  ethical  standard 
can  be  suppressed  whenever  you  think  it  will  not  approve  of  what 
you  choose  to  do. 

A  noticeable  feature  of  these  attempted  explanations  of  consecutive 
fifths  is  their  fragmentariness  and  isolation.  Most  theorists  give  only 
a  short  statement  of  what  seems  to  them  to  be  an  easy  and  obvious 
reason  for  the  prohibition  of  octaves,  namely,  the  disturbance  they 
produce  in  the  balance  or  in  the  melodic  distinctiveness  of  the  parts. 
And  some  theorists  refer  to  the  minor  restrictions  placed  upon  sequences 
of  fourths  (and  even  of  thirds)  in  confirmation  of  the  different  theory 
they  offer  for  fifths.  It  may  seem  to  many  minds  quite  satisfactory 
to  have  one  solution  for  the  octaves  and  a  second  for  the  fifths  and 
other  intervals.  Difficulties  that  are  allowed  to  slumber,  of  course 
make  no  attack.  But  there  still  remain  the  few  intervals  that  are  not 
prohibited  in  succession  at  all.  No  one  who  has  a  keen  sense  for  the 
systematic  logic  of  a  theory  can  long  remain  satisfied  with  such  work. 
And  so  the  problem  of  consecutive  fifths  remains  to-day  without  any 
recognised  solution. 

Once  the  psychical  ground  of  the  phenomena  of  music  has  been 
recognibed,  it  may  seem  to  be  an  inevitable  consequence  that  no  satis- 
factory or  convincing  explanation  of  such  phenomena  can  be  given. 
There  are  many  who  think  the  appeal  to  the  subjective  judgment 
necessarily  unconvincing.  In  his  introduction  to  his  account  of  Rameau's 
doctrines  D'Alembert  wrote  : 

Here  must  not  be  sought  that  striking  evidence  that  is  pecuhar  to  works  of 
geometry  and  that  is  so  seldom  met  with  in  those  in  which  physics  mingles.  There 
will  always  enter  into  the  theory  of  musical  phenomena  a  sort  of  metaphysics  that 
these  phenomena  impUcitly  suggest  and  that  brings  thither  its  own  natural  obscurity; 
in  this  matter  we  must  not  look  for  what  is  called  demonstration  ;  it  is  much  to  have 

7—2 


100  THE  SYSTEM  OF  FACTS  [ch. 

reduced  the  principal  facts  to  a  system  well  linked  and  well  pursued,  to  have  deduced 
them  from  a  single  experiment,  and  to  have  established  on  this  so  simple  basis  the 
best  known  rules  of  musical  art.  But,  on  the  other  hand,  if  it  is  unjust  to  exact 
here  that  intimate  and  unassailable  persuasion  that  is  produced  only  by  the  most 
vivid  light,  at  the  same  time  we  doubt  if  it  is  possible  to  throw  a  greater  light  upon 
these  matters  (9,  xiii  f.). 

Since  D'Alembert's  time  even  the  demonstrations  of  physics,  that 
have  become  so  numerous  as  to  be  a  sort  of  standard  for  all  sciences, 
have  been  subjected  to  such  searching  examination  as  to  make  some 
minds  incline  to  see  in  them  only  a  complete  degcription  of  events. 
No  doubt  there  is  much  more  involved  in  them  than  this.  But  that 
more  is  itself  the  source,  not  of  a  superiority  of  physics  to  the  science 
of  the  foundations  of  music,  but  on  the  contrary  of  a  kind  of  philosophical 
inferiority.  For  physics  imphes  the  positing  of  many  types  of  real 
entities — the  substantial  basis  of  the  phenomena  that  are  so  perfectly 
described.  At  least  all  but  a  very  few  thinkers  allow  this  feature  to 
enter  freely  into  their  physical  constructions.  Only  a  few  extremists — 
shall  we  say? — such  as  Ernst  Mach  and  Bertrand  Russell,  have  attempted 
to  claim  that  physics  as  a  science  may  be  construed  without  any  such 
postulations,  but  merely  by  description  or  by  classification  (of  pheno- 
mena) as  a  fundamental  process. 

The  science  of  music,  however,  has  as  it  were  its  whole  perspective 
in  converse  form.  Its  facts  are  obviously  phenomenal;  it  not  only  begins 
with  the  completest  description,  but,  in  the  opinion  of  many,  it  neces- 
sarily ends  there  too.  For  music,  they  hold,  is  entirely  phenomenal. 
Of  course,  when  the  basis  of  explanation  is  carried  back  into  the  physical 
realm,  as — after  sufficient  description  and  explanation  of  the  phenomena 
themselves — it  properly  may,  our  knowledge  of  the  basis  of  music 
then  goes  beyond  the  bounds  of  the  phenomenal  realm.  That,  however, 
is  not  the  point  at  issue.  Those  who  say  the  basis  of  music  is  entirely 
phenomenal  mean  that  its  whole  task  is  necessarily  mere  description. 
Description  and  classification,  and  the  study  of  sequence  and  of 
dependence  amongst  phenomena  cannot,  they  think,  lead  to  any  know- 
ledge of  phenomena  that  could  show  them  to  be  not  wholly  phenomenal. 

Only  very  few  venture  to  claim  that  the  science  of  musical  phenomena 
may  gain  knowledge  of  these  phenomena  that  shows  them  to  be  more 
than  phenomena,  to  be  at  least  partly  real,  entities  independent  of  our 
minds  that  do  not  necessarily  reveal  themselves  completely  and  finally 
to  us  at  the  first  glance  or  after  any  amount  of  inspection.  In  so  far 
then,  as  the  science  of  music  passes  so  rarely  or  never  over  the  frontiers 
of  the  phenomenal,  its  descriptions  can  proceed  with  fewer  questionable 


XII]  REGARDING  CONSECUTIVES  101 

or  uncertain  assumptions,  and  may  be  looked  upon  as  more  completely 
defensible  in  a  logical  sense  than  can  even  the  work  of  so  highly  successful 
a  science  as  physics. 

However  that  may  be,  there  is  no  doubt  at  all  nowadays  that  the 
science  of  phenomena  can  attain  to  as  complete  description  of  its  objects 
as  any  science,  and  can  thereby  compel  conviction  as  completely.  No 
doubt  the  differences  to  be  described  are  often  very  subtle;  but  they 
may  often  be  clear  and  easy  to  distinguish.  In  any  case  we  must  not 
nourish  false  expectations.  Phenomena  cannot,  for  example,  be  magni- 
fied with  microscopes.  We  are  at  the  very  outset  already  at  the  limits 
of  possible  magnification.  But  apart  from  this  sort  of  thing,  the  methods 
of  a  science  of  phenomena  are  as  reliable  and — to  those  whose  minds 
are  open  to  conviction — as  convincing  as  are  those  of  any  science  of 
nature. 

There  are  indeed  very  many  at  the  present  time  whose  minds  for 
various  reasons  have  closed  completely  against  this  idea  that  a  sufficient 
science  could  be  made  of  mere  phenomena  such  as  is  the  heard  stuff 
of  music.  They  would  not  deny  that  an  art  could  be  raised  upon  this 
basis.  But  art  they  may  feel  as  subjective  and  variable  with  the  caprice 
and  inclination  or  even  with  the  '  personality '  of  each  man.  Nevertheless 
we  must  insist  upon  it  that  the  more  an  art  is  studied,  the  more  it  is 
felt  to  be  a  realm  of  order  and  coherence.  No  doubt  in  a  science  of  art 
we  are  dealing  with  the  finer,  more  intimate,  issues  of  events,  and  not 
so  much  with  the  great  lines  of  Nature's  efforts.  But  in  arts  there  are 
also  broad  beams  of  construction.  And  the  natural  sciences  have  all 
already  come  into  contact  with  the  subtlest  and  finest  issues  of  their 
objects,  so  that  this  difference  between  the  sciences  of  Nature  and  of 
Art  has  no  longer  any  effective  validity  with  reference  to  their  dignity 
as  systems  of  knowledge. 

The  first  task  that  presents  itself  in  every  problem  of  the  foundations 
of  music  is  to  describe  the  phenomena  as  completely  as  possible. 
Preliminary  to  this  is  the  effort  to  get  all  the  phenomena  together.  That 
must  be  done  by  starting  from  the  phenomenon  that  first  raised  the 
problem, — for  example,  our  problem  of  the  prohibition  of  consecutive 
fifths, — by  searching  for  all  phenomena  in  any  degree  similar  to  this 
striking  one,  and  by  endeavouring  to  find  a  systematic  arrangement 
and  description  that  will  incorporate  them  all. 

That  is  the  logical  status  of  the  method  of  solution.  It  may  in 
some  cases  be  the  method  of  discovery  as  well.  As  a  matter  of  fact  the 


102  "  THE  SYSTEM  OF  FACTS  [ch. 

systematic  arrangement  now  to  be  expounded  was  first  suggested  by  an 
attempt  to  reduce  the  rules  for  part- writing  given  in  E.  Prout's  Harmony 
(52)  to  comprehensive  and  facile  form  by  making  a  table  of  the  objects 
shown  by  the  rules  to  be  of  chief  importance — octaves,  fifths,  fourths, 
sevenths,  seconds,  and  ninths — along  with  the  recurrent  factors  that 
seemed  to  modify  their  admission  or  prohibition, — e.g.  the  different 
voices,  the  kind  of  motion,  etc.  This  effort  suggested  its  own  extension 
and  completion  to  what  seems  highly  probable  as  at  least  a  close 
approximation  to  the  system  of  facts  of  which  consecutive  fifths  form 
a  part  and  from  which  a  satisfactory  explanation  of  their  prohibition 
may  flow.  This  system  of  facts  seems,  moreover,  to  fall  into  place  as 
an  extension  of  the  system  of  facts  regarding  the  foundations  of  music 
already  expounded.  Not  only  so  but  it  seems  also  to  renew  their  ground 
in  an  independent  manner,  which  may  lend  great  weight  both  to  the 
facts  as  already  described  and  to  the  systematic  description  and  explana- 
tion they  have  received. 

Thus  the  series  of  intervals  just  mentioned  may  be  completed  by 
the  addition  of  thirds  and  sixths.  Then  we  have  octaves,  sevenths, 
sixths,  fifth,  tritone  as  diminished  fifth  or  as  augmented  fourth,  fourth, 
thirds,  and  seconds.  We  shall  omit  consideration  of  the  so-called  prime 
or  of  unison  for  the  present  (cf.  below,  p.  112).  Th6se  are  all  the  intervals 
smaller  than  the  octave.  If  we  arrange  them  in  their  order  of  fusion, 
we  get :  octave,  fifth,  fourth,  thirds  and  sixths,  tritone,  sevenths  and 
seconds.  For  the  study  of  consecutives  in  general  this  order  is  of  much 
greater  importance  than  is  the  former. 

Of  the  circumstances  that  modify  the  prohibition  of  consecutives 
the  series  of  voices  is  one  of  the  most  important.  Let  us  begin  with 
the  consideration  of  four-part  harmony.  The  bass,  as  we  have  already 
seen,  is  the  naturally  predominant  voice.  The  next  in  order  is  the 
soprano,  partly  because  it  is  the  other  outside  voice,  and  partly  because 
it  usually  bears  the  most  important,  if  not  the  only  (coherent  or  thema- 
tised)  melody  in  harmonic  music.  It  is  the  only  voice  that  is  often  claimed 
to  be  more  noticeable  in  a  single  stationary  chord  than  is  the  bass 
(cf.  p.  51  ff.,  above).  There  is  no  obvious  distinction  in  importance  be- 
tween the  two  other  voices.  This  grading  of  the  voices  is  confirmed  and 
greatly  strengthened  by  the  way  in  which  the  stringency  of  the  rules 
of  part- writing  is  relaxed  on  occasion  in  relation  to  the  different  voices. 

But  each  interval  must  necessarily  involve  two  voices  at  once,  so 
that  the  series — bass,  soprano,  tenor  or  alto, — has  to  be  squared  with 
itself,  so  to  speak.  The  grading  that  ensues  is  :  (1)  bass-soprano,  (2)  bass- 


XIl] 


REGARDING  CONSECUTIVES 


103 


alto  or  bass-tenor,  (3)  poprano-alto  or  soprano- tenor,  (4)  alto- tenor; 
or,  as  we  shall  often  find  it  convenient  to  write  them  :  B-S,  B-A  and 
B-T,  S-A  and  S-T,  A-T. 

When  this  series  is  correlated  with  the  series  of  intervals  just  stated, 
Table  I  results.  No  definite  preconceived  idea  determined  the  form  of 
it.  The  idea  was  merely  to  arrange  the  chief  objects  referred  to  in  the 
rules  of  part- writing  as  given  by  E.  Prout  (52,  25ff.)  and  their  chief 
relations  so  that  any  system  implicit  in  them  might  become  patent. 


Table  I 
Consecufives  {preliminary  system) 

Showing  relations  between  (1)  the  stringency  of  prohibition  (Forb.  -  =  almost 
strictly  forbidden;  forb.  =  forbidden  with  exceptions;  forb.  -  =with  more  exceptions; 
+  =  allowed)  and  (2)  the  grade  of  fusion  or  consonance,  and  (3)  the  prominence 
of  the  voice-parts.   Based  upon  the  formulations  of  E.  Prout  (52,  sob.). 


Octaves 

Tritone 

to 

fifth 

Fifths 

Fourths. 

Tritone  to 

fourth 

Thirds 

or 
sixths 

Sevenths 

Seconds 

or 
ninths 

B-S 

Forb.  - 

Forb. 

forb. 

forb. 

+ 

forb. 

Forb. 

B-A 

Forb.  - 

Forb. 

forb.  - 

forb.  - 

+ 

forb. 

Forb. 

B-T 

Forb.  - 

Forb. 

forb.  - 

forb.  - 

■     + 

forb. 

Forb, 

S-A 

Forb.  - 

forb. 

forb.  - 

+ 

+ 

forb. 

Forb. 

S-T 

Forb.  - 

forb. 

forb.  - 

+ 

+ 

forb. 

Forb. 

A-T 

Forb.  - 

forb. 

forb.  - 

+ 

+ 

forb. 

Forb. 

Tritone  is  the  diminished  fifth  or  augmented  fourth,  as  the  case  may  be;  it  is 
not  reckoned  as  a  fourth  or  a  fifth  when  it  follows  a  fourth  or  a  fifth  respectively. 
The  Table  has  been  arranged  so  as  to  bring  out  a  grading  in  the  stringency  of 
prohibition  from  below  upwards  and  from  side  to  side. 

The  rules  upon  which  this  Table  is  based  are  the  following  (52,  25fl.)  : 

(1)  No  two  parts  in  harmony  may  move  [in  unison,  or]  in  octaves  with  one 
another.  There  is  one  exception  to  the  prohibition  of  consecutive  octaves.  They 
are  allowed  by  contrary  motion  between  the  primary  chords  [tonic,  dominant, 
sub-dominant]  of  the  key,  provided  that  one  part  leaps  a  fourth  and  the  other  a 
fifth. 

(2)  Consecutive  perfect  fifths  by  similar  motion  are  not  allowed  between  any 
two  parts.  They  are,  however,  much  less  objectionable  when  taken  by  contrary 
motion,  especially  if  one  of  the  parts  be  a  middle  ptirt  and  the  progression  be  between 
primary  chords  [T.D.Sd,].    This  rule  is  much  more  frequently  broken  by  great 


104  THE  SYSTEM  OF  FACTS  [ch. 

composers  than  the  rule  prohibiting  consecutive  octaves.  Consecutive  fifths  between 
the  tonic  and  dominant  chords  are  not  infrequently  met  with. 

If  one  of  the  two  fifths  is  diminished,  the  rule  does  not  apply,  provided  the 
perfect  fifth  comes  first....  But  a  diminished  fifth  followed  by  a  perfect  fifth  is 
forbidden  between  the  bass  and  any  upper  part  but  allowed  between  two  upper 
or  middle  parts,  provided  the  lower  or  occasionally  the  upper  part  moves  a  semitone. 

(3)  Consecutive  fourths  between  the  bass  and  an  upper  part  are  forbidden, 
except  when  the  second  of  the  two  is  a  part  of  a  fundamental  discord  [whose  intervals 
are  a  major  third,  a  perfect  fifth,  and  a  minor  seventh  from  the  generator^  (39,  94)] 
or  a  passing  note, — i.e.  a  note  not  belonging  to  the  harmony.  Between  any  of  the 
upper  parts  consecutive  fourths  are  not  prohibited.  They  are  sometimes  foimd 
between  the  bass  and  a  middle  part ;  but  even  these  are  not  advisable. 

(4)  Consecutive  seconds,  sevenths  and  ninths  are  forbidden  between  any  two 
parts,  unless  one  of  the  notes  be  a  passing  note....  There  is  one  important  exception 
to  this  rule  to  be  found  in  the  works  of  the  old  masters.  Corelh,  Handel,  and  others 
sometimes  followed  a  dominant  seventh  by  another  seventh  on  the  bass  note  next 
below. 

There  is  some  difference  of  opinion  amongst  authorities  as  to  the 
special  digressions  from  the  prohibitions  that  are  admissible.  But  if 
we  take  account  chiefly  of  the  existence  and  degree  of  freedom  of 
exceptions  we  may  look  upon  Front's  rules  as  relatively  valid. 

Thus  Parry  (45)  says  that  "there  are  so  many  consecutive  sevenths 
to  be  found  in  the  works  of  the  greatest  masters,  and  that,  when  they 
are  harsh,  they  are  so  obviously  so,  that  the  rule  prohibiting  them 
seems  both  doubtful  and  unnecessary."  Here  the  point  of  view  is 
mainly  practical.  The  ugliness  of  the  sequence  is  really  admitted, 
although  in  a  restricted  form;  and  so  it  appears  in  our  Table.  Text- 
books of  harmony  evidently  find  it  unnecessary  to  state  that  consecutive 
thirds  and  sixths  are  unobjectionable.  But  the  fact  is  of  the  greatest 
importance  as  a  datum  for  theoretical  work  for  all  that.  It  is  just  these 
obvious  facts  that  no  one  mentions  that  are  often  the  keystone  of  a 
successful  theory  of  such  phenomena. 

There  is  evidently  a  system  inherent  in  the  Table.  For  we  see  that 
the  two  outside  columns  contain  nothing  but  Forb.  or  Forb.  —  ,  i.e. 
strict  or  almost  strict  prohibitions  for  all  voices.  In  the  second  column 
the  prohibition  is  relaxed  a  little  in  the  lower  three  pairs  of  voices 
(forb.).  In  the  third  column  that  relaxation,  and  perhaps  even  a  little 
more  of  it  (forb.  —  ),  holds  for  all  the  pairs  of  voices  except  B-S.  In 
the  fourth  column  further  progress  is  made  in  the  same  direction;  the 
sequence  is  admitted  for  the  lower  three  pairs  of  voices.    In  the  case 

^  The  'generator'  is  the  lowest  note  of  the  so-called  root-position  of  the  chord. 


xn]  REGARDING  CONSECUTIVES  105 

of  the  thirds  and  sixths  there  is  perfect  freedom.  But  the  prohibition 
comes  into  force  again  in  a  restricted  degree  for  the  sevenths,  and  is 
finally  complete  for  the  seconds  and  ninths.  The  grading  of  the  voice- 
pairs  that  makes  this  system  possible  is  identical  with  the  grading 
deduced  above  from  the  experimental  and  general  facts  of  analysis 
of  chords. 

The  most  significant  and  suggestive  feature  of  the  Table,  however, 
is  the  sequence  of  the  intervals  seen  in  the  top  horizontal  column. 
This  sequence  is  very  nearly  the  same  as  that  already  given  for  the 
grades  of  consonance  of  intervals  that  lie  within  the  octave.  And  the 
result  suggests  strongly, — so  far  at  least  as  this  Table  shows  the  situa- 
tion.— that  the  degree  of  stringency  of  the  prohibition  of  consecutive 
intervals  of  the  same  species  depends  (1)  upon  the  grade  of  consonance 
or  dissonance,  and  (2)  upon  the  prominence  of  the  two  voices  that 
constitute  the  intervals  in  question.  The  Table  thus  takes  proper 
notice  of  the  fact  that  the  consecutive  intervals  must  both  lie  between 
the  same  two  voices. 

A  special  problem  is  created  in  the  Table  by  the  tritone.  As  a 
so-called  diminished  fifth  it  is  not  forbidden  when  it  follows,  but  only 
when  it  precedes,  a  perfect  fifth.  The  same  holds  in  so  far  as  it  is  reckoned 
as  an  Augmented  fourth  in  connexion  with  a  perfect  fourth.  The  prohibi- 
tion of  the  tritone  when  the  fifth  follows  it,  seems  to  be  stricter  even 
than  the  prohibition  of  two  fifths.  This  extreme  difference  between  the 
two  successions  shows  that  we  are  here  not  dealing  with  consecutive 
intervals  of  the  same  species.  When  the  augmented  fourth  or  diminished 
fifth  follows,  it  is  not  a  fifth  or  a  fourth  at  all,  but  another  kind  of 
interval — at  least  as  far  as  the  system  of  facts  represented  in  the  Table 
is  concerned.  It  is  for  that  reason  it  has  been  classified  in  the  Table 
as  a  tritone.  Thus  the  problem  of  the  tritone  reduces  itself  to  the  single 
case  in  which  it  precedes  either  a  fifth  or  a  fourth.  This  is  obviously 
not  an  instance  of  consecutive  intervals  of  the  same  kind.  In  respect 
of  the  fifth  alone  it  belongs  to  the  case  of  'hidden  fifths.'  Otherwise 
the  succession  of  tritone  and  fifth  belongs  to  the  class  of  problems 
that  includes  all  such  questions  as  :  under  what  circumstances  may 
any  two  intervals  of  different  species  follow  one  another  in  the  same 
voices? 

We  may,  therefore,  remove  the  tritone  from  its  present  position 
in  the  preliminary  Table  of  consecutives  and  replace  it  properly,  having 
regard  solely  to  the  degree  in  which  consecutive  tritones  are  avoided 


106  THE  SYSTEM  OF  FACTS  [ch. 

or  forbidden  in  part-writing.  Text-books  of  harmony  contain  no  state- 
ment regarding  consecutive  tritones,  either  as  diminished  fifths  or  as 
augmented  foiirths.  But  in  dealing  with  modulation  a  practical  oppor- 
tunity occurs  of  presenting  information  on  this  subject.  Thus  P. 
Tchaikovsky  says  (76,  70  f.): 

There  are  also  sequences  in  which  every  chord  constitutes  a  modulation.  They 
are  those  in  which  dominant  seventh  chords  or  other  chords  resolving  into  the 
tonic  succeed  one  another,  always  falling  a  fifth  or  rising  a  fourth,  as  in  a  sequence 
within  the  limits  of  one  key.  In  such  a  sequence  each  chord  resolves  into  a  chord 
which  itself  demands  resolution  and  forms  at  the  same  time  the  resolution  of  its 
precursor. 

He  then  gives  progressions  containing  five  to  eight  tritones  in 
succession,  diminished  fifths  alternating  with  augmented  fourths.  Of 
course  there  is  some  difference  between  these  two  intervals  in  their 
musical  significance.  But  in  respect  of  their  fusion  and  even  in  respect 
of  their  specific  nature  as  intervals  (proportions  of  volumes)  there  is 
practically  none.  In  one  of  Tchaikovsky's  examples  there  is  even  a 
series  of  eight  simultaneous  fairs  of  tritones  (chords  of  the  diminished 
seventh),  one  tritone  lying  between  the  two  outer  voices,  the  other 
between  the  two  inner  voices.  Prout  (52,  1st  ed.,  162)  gives  an  example 
from  Bach'is  Chromatic  Fantasia  in  which  successive  tritones  abound. 

We  may,  therefore,  look  upon  successive  tritones  as  being-  more 
freely  admissible  than  consecutive  sevenths,  major  or  minor.  The 
diminished  seventh,  which  may  be  run  in  succession  (52,242)  is  practically, 
and  from  the  point  of  view  of  fusion,  quite  the  same  thing  as  the  major 
sixth,  which  is  not  restricted  at  all.  No  doubt  the  musical  afiSnities  of 
intervals  that  are  apprehended  as  diminished  sevenths  will  call  for  a 
different  treatment  of  them  from  that  of  intervals  apprehended  as  major 
sixths.  But  it  is  clear  that  in  dealing  with  consecutives  we  are  not 
concerned  with  such  special  apprehension  of  musical  setting  and  relation- 
ship, but  with  a  more  fundamental  matter  that  appertains  to  the 
intervals  in  question  almost  in  any  setting,  so  long  as  they  lie  between 
the  same  voices. 

We  may,  therefore,  place  the  tritone  in  an  amended  Table  of 
Consecutives  between  the  thirds,  sixths  and  the  sevenths. 

This  brings  the  final  Table  into  much  greater  conformity  with  the 
experimentally  established  grading  of  fusion  of  the  different  intervals. 
In  fact  so  far  as  the  differentiation  of  our  Table  shows,  in  which  the 
different  thirds  or  sixths  or  sevenths,  etc.,  are  not  distinguished,  the 


xn] 


EEGARDING  CONSECUTIVES 


107 


conformity  is  complete.  The  most  frequent  grading  of  fusion  that 
experimental  research  has  as  yet  shown  is  (77, 104) :  0,  5,  4,  III,  3,  VI, 
6,  T,  II,  7,  2,  VII  (cf.  above,  p.  16). 

Table  II 
Consecutives  {Final  System) 

Showing  relations  between  (1)  the  stringency  of  prohibition  (Forb.,  Forb.  -, 
forb.,  forb.  - ,  +  or  allowed),  and  (2)  the  grade  of  fusion  of  any  interval,  and  (3)  the 
prominence  of  the  voice-parts  in  which  the  interval  appears. 


O's 

6's 

4's 

3'8  &  6'8 

T's 

7's 

2's  &  9's 

B-S 

Forb.  - 

forb. 

forb. 

+ 

forb.  - 

forb. 

Forb. 

B-A 

Forb.  - 

forb.  - 

forb.  - 

+ 

forb.  - 

forb. 

Forb. 

B-T 

Forb.  - 

forb.  - 

forb.  - 

+ 

forb.  - 

forb. 

Forb. 

S-A 

Forb.  - 

forb.  - 

+ 

+ 

forb.  - 

forb. 

Forb. 

S-T 

Forb.  - 

forb.  - 

+ 

+ 

forb.  - 

forb. 

Forb. 

A-T 

Forb.  - 

forb.  - 

+ 

+ 

forb.  - 

forb. 

Forb. 

The  Table  shows  that  the  grades  of  fusion  from  greatest  consonance  to  greatest 
dissonance — in  relation  with  the  relative  prominence  of  the  pair  of  voices  on  which 
the  interval  in  question  rests — give  rise  to  a  system  of  preferences  or  prohibitions 
of  a  very  well  graded  kind  (cf.  p.  103,  above). 

And  the  following  conclusion  may  be  drawn.  If  due  consideration 
is  given  to  the  prominence  of  the  pair  of  voices  that  bear  the  interval 
in  question,  it  appears  that  the  immediate  repetition  of  an  interval 
in  the  same  voices  is  the  more  offensive  the  greater  the  consonance  or 
dissonance  of  that  interval.  The  point  of  minimal  unpleasantness  or 
of  maximal  pleasantness  (as  the  case  may  be)  in  the  series  from  greatest 
consonance  to  greatest  dissonance  lies  amongst  the  thirds  and  sixths. 
These  intervals  may,  therefore,  be  held  to  be  fusionally  neutral. 

This  inference  differs  a  little  from  the  prevalent  attitude  towards 
the  thirds  and  sixths.  They  are  nowadays  ranked  among  the  consonances. 
They  are,  of  course,  certainly  not  dissonances.  But,  on  the  other  hand, 
they  are  perhaps  not  really  consonances  either. 

It  is  a  familiar  fact  that  the  ancient  Greeks  did  not  include  them 
amongst  their  consonances,  which  were  octave,  fifth,  and  fourth,  alone, 
and  stated  in  this  order  by  Aristoxenus  and  by  Ptolemy  (cf.  66,  38,  58). 
This  fact  has  often  been  interpreted  as  indicating  that  the  Greeks 
considered  the  thirds  and  sixths  to  be  dissonances,  as  we  now  understand 
this  term.  But  that  may  not  be  taken  for  granted.  The  system  indicated 


108  FACTS  REGARDING  CONSECUTIVES  [ch.  xii 

in  Table  II  suggests  strongly  that  the  thirds  and  sixths  may  not  have 
been  included  amongst  the  consonances  by  the  Greeks  because  they 
are  not  appreciably  'positive  degrees  of  consonance^.  That  we  find  them 
highly  pleasant  and  characteristic  is  not  at  ail  inconsistent  with  the 
correctness  of  this  estimate. 

Of  course  the  mere  fact  that  we  rank  the  thirds  and  sixths  after 
the  fourth  in  the  grading  of  fusions  that  lead  from  greatest  consonance 
to  greatest  dissonance,  implies  nothing  at  all  as  to  whether  these  intervals 
are  consonances  or  dissonances.  And  the  experimental  evidence 
regarding  the  very  slight  percentage  difference  between  the  grades  (of 
approximation  to  the  impression  of  a  single  tone)  lower  than  the  fourth, 
and  the  similar  evidence  regarding  the  variations  in  the  serial  arrange- 
ment of  the  grades  of  fusion  show  that  at  least  there  is  no  clear  division 
between  these  lower  grades.  So  if  a  minor  seventh  is  a  dissonance,  a 
minor  third  can  hardly  be  a  strong  consonance;  nor  can  even  a  major 
third.  There  must  be  a  point  at  which  dissonance  passes  into  con- 
sonance. Logically  that  point  may  be  a  vanishing  point,  of  course. 
But  even  then  the  lower  consonances, — if  we  suppose  the  thirds  and 
sixths  to  be  positive  consonances, — must  have  a  very  low  degree  of 
consonance  to  be  so  slightly  different  from  the  lesser  dissonances  and 
so  often  confusible  with  them  in  respect  of  fusion.  It,  therefore,  seems 
probable  that  the  grades  of  fusion  including  the  thirds  and  sixths  may 
properly  be  considered  to  be  neutral. 

Thirds  and  sixths,  then,  are  neither  distinct  dissonances,  nor  are 
they  distinct  consonances.  And  the  Table  of  Consecutives  gives  us  a 
very  strong  reason  for  accepting  this  description.  For  the  treatment 
there  shown  to  be  accorded  to  thirds  and  sixths  is  distinctly  different 
from  that  accorded  to  the  extreme  consonances  and  to  the  extreme 
dissonances. 

^  Cf.  Gevaert  (14,  102):  "Let  us  notice  first  that  the  meaning  of  the  terms  has  been 
modified  in  the  course  of  time.  We  translate  symphonia  by  consonance,  diaphonia  by 
dissonance.  So  did  even  the  Romans  in  the  Augustan  age,  always  attaching  to  these 
words  another  idea  than  we  do.  The  fundamental  difference  distinguished  by  the  ancients 
between  the  two  kinds  of  intervals  is  that  in  symphony  the  sounds  fuse  to  a  perfect  unity, 
whilst  in  diaphony  they  maintain  their  individuality  and  detach  themselves  in  some  way 
from  one  another.  In  this  respect  our  impression  does  not  differ  appreciably  from  that 
of  the  Greeks.  For  us  too  the  thirds  and  sixths  have  a  clear  cut  character  that  is  lacking 
in  the  fifth  and  fourth.  We  notice  the  same  clearness  in  the  second  and  in  the  seventh; 
and  from  this  new  point  of  view  we  find  it  possible  to  let  the  ranking  of  these  two  intervals 
in  the  same  category  as  the  thirds  and  sixths  pass."  Gaudentius  was  the  first  to  admit 
the  major  third  (and  also  the  tritone)  amongst  the  consonances.  To  the  former  inclusion 
we  now  generally  agree,  but  only  with  special  effort  to  the  latter  (cf.  14,  99;  66,  7i  f.). 


CHAPTER  XIII 

THE  REASON  FOR  THE  PROHIBITION  OF  CONSECUTIVES 

The  question  which  next  arises  is  the  one  from  which  all  previous 
writers  on  the  subject  have  started  :  why  are  consecutives  offensive? 
Why  are  all  these  consecutives  offensive,  each  in  its  degree? 

The  system  of  facts  we  have  discovered  in  the  preceding  chapter 
does  not  answer  the  question  directly.  It  only  arranges  the  objective 
facts  with  which  any  answer  to  the  question  must  reckon.  It  indicates 
that  the  solution  must  not  only  be  the  same  for  octaves  as  for  fifths, 
but  it  must  even  be  the  same  for  both  dissonances  and  consonances. 
The  only  thing  common  to  them  all  is  some  degree  of  fusion.  The 
repetition  of  a  high  grade  consonance  or  dissonance  introduces  a  new 
and  special  feature  that  is  unpleasant.  Thus  our  task  must  now  be  to 
show  a  basis  for  this  feature  and  to  form  a  theory  as  to  its  nature  which 
will  adequately  justify  on  conceptual  grounds  the  unpleasantness  of 
the  effect  we  hear. 

The  problem  may  be  approached  in  two  ways.  In  the  first,  experi- 
mentally, we  might  present  a  series  of  observers  with  a  systematically 
varied  complex  of  consecutive  pairs,  and  ask  for  direct  observation  and 
description  of  the  feature  of  each  that  is  unpleasant  or  more  or  less 
preferable.  This  course  would  certainly  not  be  successful  in  such  a 
simple  form,  although  it  would  be  of  great  value  for  the  grading  of  the 
intervals  on  the  basis  of  their  preferability.  Consecutive  intervals  of 
the  same  species  have  been  considered  and  compared  and  reflected 
upon  for  centuries  already  without  even  any  indication  of  agreement 
having  been  reached  as  to  what  it  is  in  the  sequence  that  is  directly 
unpleasant.  Even  those  who  have  written  treatises  on  the  subject 
have  hardly  done  more  than  guesswork  upon  the  problem,  except  in 
so  far  as  they  attempted  to  infer  a  basis  of  unpleasantness  from  the 
system  of  facts  gathered  round  the  central  object  of  inquiry. 

The  situation  in  this  particular  aspect  of  it  is  similar  to  that  of  the 
theory  of  consonance  and  dissonance.  In  spite  of  the  fact  that  the 
ancient  Greeks  and  the  older  writers  of  the  modern  era  had  defined 
consonance  as  the  mixture  or  blending  of  two  tones  into  one,  that 
direct  description  did  not  receive  in  the  more  modern  explanations  by 
the  relations  of  the  harmonics  the  central  importance  due  to  it.   It  was 


110  THE  REASON  FOR  [ch. 

only  restored  to  its  proper  position  by  the  critical  studies  of  Stumpf. 
And  even  Stumpf  could  not  go  beyond  this  amount  of  direct  description, 
already  attained  by  the  Greeks,  to  say  more  definitely  and  decisively 
how  the  fusing  tones  interpenetrated  one  another  so  as  to  approximate 
to  the  effect  of  a  single  tone.  If  it  was  almost  impossible  in  this  case 
to  proceed  beyond  the  terms  of  direct  description  to  an  adequate  theory 
of  the  basis  of  the  phenomenon  in  the  sensory  stuff  of  the  tones  them- 
selves, how  can  we  expect  by  direct  observation  to  win  a  theory  of  the 
bad  effect  of  consecutives,  seeing  that  even  the  direct  description  of 
that  effect  has  not  yet  been  obtained? 

Evidently  the  work  of  observation  and  description  must  be  facilitated 
by  the  discovery  of  definite  alternative  questions,  to  be  answered  by 
the  comparison  of  minor  differences.  In  other  words,  we  must  learn 
how  to  instruct  the  observer  so  as  to  make  description  easier  for  him 
by  directing  his  attention  precisely  towards  possible  special  features 
of  consecutives.  We  must  expect  a  properly  instructed  and  careful 
course  of  systematic  observation  to  confirm  any  inferences  as  to  the 
basis  of  the  prohibitions  that  may  otherwise  be  gathered. 

For  that  is  the  other  way  of  approaching  the  problem.  By  enlarging 
the  system  of  facts  in  which  consecutive  fifths  stand,  we  have  already 
obtained  a  much  better  formula  for  their  prohibition  than  we  could 
have  obtained  from  a  study  of  them  alone.  The  fifths  are  forbidden 
because  of  something  that  emerges  from  pairs  of  highly  positive  or 
negative  fusions.  Perhaps  if  we  enlarge  in  turn  the  system  of  facts 
of  which  consecutives  form  a  part,  we  may  attain  some  still  more 
specific  formula.  Armed  with  this,  we  could  return  to  the  work  of 
direct  description  with  some  hope  of  obtaining  a  definite  answer  to  a 
definite  question.  A  probable  further  system  of  facts  suggests  itself 
in  the  well-known  counterpart  to  consecutives — the  prohibition  of 
single  intervals,  commonly  termed  hidden  octaves  and  fifths. 

But  in  the  facts  already  before  us  there  is  an  aspect  that  calls  for 
some  notice,  although  it  may  at  first  glance  seem  trivial  and  obvious  : 
that  the  intervals  to  be  prohibited  must  lie  between  the  same  voices. 
Of  course  that  is  not  equivalent  to  saying  that  they  must  lie  somewhere; 
they  might  lie  between  different  voices,  and  when  they  do  so,  they  give 
rise  to  no  feature  that  is  objectionable.  Evidently  when  we  listen  to 
music  in  several  parts,  our  attention — even  in  music  that  is  pre- 
dominantly harmonic — runs  along  the  voices,  as  it  were,  noticing  the 
series  of  relations  that  emerge  between  the  successive  tones  of  each 


XIII]  THE  PROHIBITION  OF  CONSECUTIVES  111 

pair  of  voices.  It  does  not  connect  into  systems  one  relation  between 
one  pair  of  voices,  a  second  relation  between  another  pair  of  voices, 
a  third  relation  between  a  third  pair,  and  so  on^.  The  systems  are  rather 
those  that  actually  present  themselves  serially.  The  relations  in  question, 
however,  are  fusional,  or,  as  Hullah  said,  perpendicular  relations.  Only, 
chords  are  not  apprehended — even  in  music  that  is  predominantly 
harmonic — as  unanalysed  wholes;  the  apprehension  is  not  fusional  or 
perpendicular  throughout  the  chord  as  an  undivided  unity.  Analysis 
breaks  this  whole  into  parts  of  two  voices  at  least;  these  are  the  units  of 
fusional  or  perpendicular  apprehension. 

In  polyphonic  music  this  much  is  also  undoubtedly  true.  Only  here 
there  enters  another  factor  that  justifies  the  contrary  term  'horizontal 
apprehension,'  namely  the  distinctively  melodic  or  thematic  treatment 
of  each  voice.  The  figures,  forms,  and  phrases  of  melody  are  maintained 
in  each  voice  over  and  above  the  restrictions  that  are  placed  upon 
their  progressions  by  the  fusional  aspects  of  pairs  of  voices.  It  is  not, 
then,  so  much  the  case  that  harmonic  music  has  introduced  a  feature 
not  yet  present  in  polyphonic  music;  but  rather  in  the  latter  there  is 
present  in  highly  cultivated  form  a  feature  which  prevents  the  harmonic 
relations  implicit  within  it  from  coming  into  prominence.  No  doubt, 
too,  this  suppression  prevented  these  relations  from  being  specially 
cultivated.  But,  whether  cultivated  or  not,  they  are  essentially  present 
in  both  types  of  music. 

Thus  we  obtain  some  closer  specification  of  the  relations  between 
synthesis  and  analysis  in  music  generally.  In  libtening  to  music  in  several 
parts  we  do  not  apprehend  the  fusions  of  chords  in  so  far  as  they 
approximate  to  the  balance  and  symmetry  of  a  single  tone  as  a  whole 
mass.  Our  attention  is  always,  up  to  a  certain  degree,  analytic.  We 
notice  always  the  relations  between  pairs  of  voices.  And  to  do  so  we 
must  be  able  to  maintain  the  proportions  of  the  volumes  as  defined  by 
each  pair  of  voices  in  the  forefront  of  our  attention.  For  that  purpose 
analysis  is  necessary. 

Now  we  have  a  perspective  from  which  to  judge  the  generalisation 
attained  from  Table  II.  A  succestion  of  high  grade  consonances  or 
dissonances  is  very  unpleasant;  it  is  offensive  according  to  the  degree 
of  consonance  or  dissonance  (or  of  fusion  positive  to  the  neutral  grades 

^  So  the  crossing  of  parts  will  obviate  consecutive  fifths  that  appear  when  the  parts  are 
in  pianoforte  score  (cf.  52,  im  f.  for  example).  But  such  voice-leading  will,  of  course,  require 
the  support  of  a  difference  in  blend  between  the  voices,  as  in  choral  or  chamber  music. 


112  THE  KEASON  FOR  [ch. 

or  negative  to  them)  and  to  the  prominence  of  the  pair  of  voices  con- 
cerned. In  short,  prominence  of  high  or  low  grade  fusion  disturbs; 
neutral  grades  of  fusion  do  not  disturb.  Disturb  what?  Only  one 
answer  suggests  itself  :  they  disturb  (the  analysis  or  the  set  of  attention 
required  to  maintain)  the  usual  flow  of  presentation  of  relations  between 
the  pairs  of  voices.  The  horizontal  view,  so  far  as  it  is  generally  attained 
in  music,  is  disturbed  by  the  undue  prominence  of  the  perpendicular 
relation  between  the  voices.  Either  the  voices  interpenetrate  too  much 
in  successive  pairs  so  as  to  cut  off  the  connexion  between  the  two  tones 
of  either  voice  (thus  the  connexions  c-d  and  g-a  are  broken  in  con- 
secutive fifths  on  c  and  d)\  or  the  voices  disrupt  from  one  another  too 
markedly  and  thus  also  break  the  connexion  unduly  within  each  voice. 
Neutral  grades  of  fusion  alone  do  not  in  succession  break  this  even 
flow  of  analytic  concentration  necessary  for  the  appreciation  of  the 
greater  works  of  music  ^. 

We  cannot  go  into  the  further  aspects  of  this  formulation  at  once. 
We  must  await  the  systematic  arrangement  of  the  facts  included  in 
these  further  aspects,  keeping  the  formulation  before  us  as  a  hypothesis 
to  be  tested  and  enriched.  But  it  is  at  least  evident  now  why  con- 
secutives  are  forbidden  only  in  connexion  with  a  movement  of  the  voices. 
The  repetition  of  any  interval  without  any  change  of  its  pitch  would 
in  no  way  affect  the  apprehension  of  the  sequent  tones  of  each  voice. 
For  as  nothing  has  changed,  the  attention  has  for  the  moment  an 
easier  task  than  usual.  On  the  other  hand  a  succession  of  unisons  tends 
to  betray  the  analytic  attention  into  losing  hold  of  the  individuality 
or  duple  nature  of  the  voices  that  thus  temporarily  coincide.  One 
unison,  however,  is  not  disturbing  so  long  as  the  voices  are  felt 
melodically  to  converge  and  to  coalesce ;  and  if  the  next  chord  is  suitable, 
they  will  be  felt  to  separate  and  to  diverge  again  (cf.  below,  p.  130). 

^  Descartes  explained  the  prohibition  of  consecutive  octaves  and  fifths  thus:  "Ratio 
enim  quare  id  magis  expresse  prohibeatur  in  his  consonantiis  quam  in  aliis,  est  quia  hae 
sunt  perfectissimae;  ideoque,  dum  una  ex  illis  audita  est,  tunc  plane  auditui  satisf actum 
est.  Et  nisi  illico  alia  consonantia  ejus  attentio  renovetur,  in  eo  tantum  occupatur,  ut 
advertat  parum  varietatem  et  quodammodo  frigidam  cantilenae  symphoniam.  Quod 
idem  in  tertiis  aliisque  non  accidit:  immo,  dum  illae  iterantur,  sustentatur  attentio, 
augeturque  desiderium,  quo  perfectiorem  consonantiam  expectamus"  (11,  132).  The  com- 
parison with  the  case  of  thirds  is  interesting.  But  it  serves  only  to  bring  out  the  older 
point  of  view  which  concentrated  on  the  perfect  consonances,  and  not  the  modem  point 
of  view  in  which  the  thirds  play  the  more  essential  part;.  Descarties'  theory  is  of  the 
'variety'  type.  That,  however,  is  true  only  as  an  approximation  towards  what  variety 
makes  possible  and  what  is  attainable  in  some  cases  (e.g.  with  thirds)  even  without  variety, 
namely  continmty  of  melody. 


xiii]  THE  PROHIBITION  OF  CONSECUTIVES  113 

It  must  also  be  now  clear  that  concurrence  of  voices  in  fifths  or 
fourths  is  only  tolerable  in  a  primitive  stage  of  music.  There  the  homo- 
phonic  interest  is  almost  the  only  one  present  in  the  music.  Polyphonic 
relations  have  either  not  yet  been  attained  at  all  or  only  on  rare  occasions, 
so  that  even  when  they  do  occur,  the  ready  dispositions  of  the  hearer's 
mind  will  not  easily  yield  to  any  unpleasantness  they  may  bring  when 
consecutives  appear.  Both  singers  and  hearers  intend  and  know  the 
concurrent  voices  to  be  the  same  melody.  No  doubt  they  have  to  will, 
or  to  attend  to,  the  melodic  continuity  more  energetically  when  they 
use  consecutive  fifths  and  fourths  than  when  they  use  octaves  or  a 
bare  melody.  But  they  may  be  quite  willing  to  do  so  for  the  sake  of 
the  variety  thus  attained,  until  further  variation  and  closer  attention 
show  them  that  the  bad  effects  thus  ignored  have  no  compensating 
power  to  please  by  heightening  contrast;  or  that,  if  they  have  this 
power,  the  systems  of  variation  made  possible  by  it  are  so  small  and 
weak  as  compared  with  the  systems  of  variation  admitted  by  changes 
of  consonances,  and  especially  by  the  use  of  the  lower  degrees  of  con- 
sonance, that  they  are  not  worth  while,  or  are  not  profitable  lines  of 
development,  and  so  are  best  barred  out  altogether.  Hence  their 
gradual  disappearance  from  music  as  it  progressed  towards  the  form 
and  style  of  distinct  polyphony,  and  their  vigorous  prohibition  until  it 
had  developed  enough  to  allow  of  their  re-introduction  amongst  many 
minor  systems  of  variation  in  a  way  that  does  tend  to  enrich  the 
structural  potentialities  of  music. 

On  the  other  hand  the  octave  does  not  make  melodic  continuity 
at  all  diflficult  to  maintain  so  long  as  the  presence  of  the  intention  to 
such  continuity  has  been  made  evident  or  so  long  as  the  intention 
towards  melodic  diversity  has  not  been  declared.  The  reason  for  this 
freedom  is  not  so  much  the  fact  that  the  octave  is  the  first  harmonic 
of  a  fundamental,  whereas  the  fifth  is  the  second.  For  that  should 
only  establish  a  gradation  of  difficulty,  as  it  does  perhaps  in  the  primitive 
mind,  not  a  difference  between  pleasure  and  offensiveness,  as  it  does 
in  our  music.  The  reason  is  rather  that  in  the  systems  of  intervals  of 
our  music  or  in  our  tonality  the  octave  is  the  absolute  basis  of  reference 
of  all  intervals,  and  is  so  because  of  the  fact  that  the  increase  of  an 
octave  means  the  decrease  of  volume  by  half,  and  because  this  difference 
does  not  alter  or  distort  any  pattern  of  volumic  proportions  (cf.  above 
p.  72  ff.).  A  tone  and  its  octave  are  therefore  very  easily  apprehended 
as  one  thing,  and  that  unit  of  pattern  may  be  followed  with  great  ease 
throughout  all  sorts  of  changes  of  its  volume  as  a  whole.   The  doubling 

W.  F.  M .  8 


lU    REASON  FOR  PROHIBITION  OF  CONSECUTIVES   [ch.  xiii 

of  a  melody  in  octaves,  then,  is  admissible  in  our  music  because  it  is 
quite  easy  to  follow  melodically  and  it  is  quite  consistent  with  the 
volumic  structure  of  our  systems  of  intervals. 

It  has  sometimes  been  said  that  the  reason  for  the  prohibition  of 
consecutive  octaves  was  that  the  effect  of  a  four-part  harmony  was 
thereby  lost.  It  is  now  evident  from  our  system  of  facts  and  from  the 
place  of  the  octave  in  it  that  the  reason  cannot  be  of  this  merely  negative 
order.  The  consecutive  octaves  must  present  a  big  positive  something 
that  is  offensive.  We  shall  form  a  clearer  idea  of  what  this  is  as  we 
proceed. 


CHAPTER  XIV 

EXCEPTIONS  TO  THE  PROHIBITIONS  OF  CONSECUTIVES 

In  arranging  the  system  of  facts  regarding  consecutives  we  had  to  be 
content  with  an  approximation  to  agreement  in  the  statement  of  the 
rules.  It  was  enough  to  bring  out  the  general  trend  of  the  differences 
included  within  the  system  without  striving  to  define  it  exactly  in  its 
absolute  form.  Fortunately  there  is  not  very  much  difference  of  opinion 
in  the  statements  of  these  rules  given  in  the  chief  text-books  of  harmony. 
In  arranging  the  rules  which  state  exceptions  to  the  prohibitions  of 
consecutives  we  shall  again  have  to  rely  upon  some  estimate  of  the 
main  trend.  The  general  and  growing  agreement  amongst  theorists 
will  facilitate  our  work  for  the  present. 

A  full  and  sufficient  account  of  these  rules  and  their  exceptions 
would  best  be  based  upon  a  very  extended  statistical  treatment  of  the 
musical  material.  No  doubt  some  of  the  theorists  who  have  worked 
out  rules  of  prohibition  and  of  admission  have  collected  large  numbers 
of  instances  and  have  based  their  generalisations  upon  them.  But  for 
the  fullest  understanding  much  would  be  gained  from  an  analytic 
study  and  a  statistical  manipulation  of  such  a  collection,  if  it  were 
published  in  an  extended  form,  so  that  the  reader  might  follow  the 
relative  quantitative  importance  of  the  various  factors  that  are  found 
in  groups  of  exceptions.  What  is  required  for  an  elementary  knowledge 
of  the  principles  of  construction  has  doubtless  already  been  attained. 
But  even  in  some  elementary  matters  these  formulations  have  become 
detailed  enough  to  show  considerable  divergence  of  opinion.  This 
divergence  is  possibly  not  so  much  a  sign  of  any  difference  in  aesthetic 
reaction  between  persons  or  of  any  aesthetic  differences  in  their  nature, 
as  rather  a  result  of  the  consideration  by  each  of  them  of  different 
special  groups  of  exceptions,  or  of  only  some  of  the  factors  operative 
in  typical  cases  in  abstraction  from  other  accompanying  ones  that 
contribute  to  the  final  aesthetic  effect.  In  any  case  there  can  be  little 
doubt  that  an  analytic  study  of  a  large  number  of  exceptions  on  a 
statistical  basis  would  be  of  great  service  both  to  the  science  and  to  the 
art  of  music.  This  sort  of  effort  may  be  commended  to  those  who  have 
any  favourable  opportunity  for  making  such  large  collections. 

8—2 


116  EXCEPTIONS  TO  THE  [ch. 

The  exceptions  admitted  are  few.  For  octaves  (and  fifths)  Macfarren 
(35,  82 f.)  stated  that  the  use  of  the  sequence,  "however  rare,  by  com- 
posers of  the  present  century,  proves  that  this  most  stringently 
proscribed  progression  may  produce  an  effect  of  measureless  beauty, 
when  it  lies  between  the  chord  of  the  tonic  and  either  'dominant  or 
subdominant,'  provided  only  that,  in  the  case  of  octaves,  the  parts 
that  have  the  two  in  succession  proceed  by  contrary  motion  "  (cf .  A.  Day, 
10,  58).  And  Prout,  as  quoted  above,  agreed  to  this  exception  in  the 
second  edition  of  his  work  on  harmony.  Macfarren  illustrates  the  point* 
from  Beethoven's  Pastoral  Symphony  (between  the  bass  and  the  alto) 
and  from  Sonata,  Op.  53  (major  common  chord  with  doubled  root  in 
each  hand  on  dominant  and  then  on  tonic).  Similarly  Tchaikovsky 
notes  that  in  strict  part- writing  "(fifths  and)  octaves  are  permitted 
in  the  inner  voices  if  contrary  motion  be  employed"  (76,  lis).  Parry  (45) 
points  out  that  consecutives  are  most  objectionable  in  vocal  and 
chamber  music;  in  pianoforte  and  orchestral  music  they  are  often  lost. 

Shinn  discusses  the  question  at  some  length  (58,  276f.),  and  claims 
that  the  sequence  of  octaves  in  contrary  motion  or  of  octave  and 
unison  often  produces  "an  exceptionally  strong  musical  effect."  He 
generalises  beyond  the  tonic-to-dominant  or  subdominant  relation 
stated  by  Macfarren  towards  "other  pairs  of  triads  standing  in  a  similar 
relationship  with  regard  to  their  progression  such  as  the  triads  upon 
the  mediant  and  submediant,"  etc.,  "but  these  are  not  often  employed." 
The  musical  strength  of  these  progressions  is  not  due  "entirely  to  the 
fact  of  their  harmonic  relationship  (or  root  progression),  but  partly 
also  to  the  movement  of  the  outside  parts  by  almost  equal  intervals — 
that  is,  one  by  a  fourth  and  the  other  by  a  fifth."  The  effect  when  one 
part  moves  a  third  and  the  other  a  sixth,  especially  when  the  bass 
moves  the  sixth,  is  generally  less  strong.  Consecutives  can  rarely  be 
employed  in  a  satisfactory  manner  when  one  part  moves  a  second  and 
the  other  a  seventh.  It  is  not  clear  whether  Shinn  means  the  harmonic 
relationship  of  the  chords  or  the  mere  movements  of  the  voices  to  be 
the  more  important  element  in  the  effect;  probably  the  latter. 

Octaves  in  similar  motion  are  admissible  according  to  their  purpose 
and  position.  As  examples  Shinn  gives  one  (from  Beethoven)  for  "the 
emphasising  of  a  full  cadence  by  the  outside  parts  moving  from  dominant 
to  tonic,"  another  for  the  formation  of  a  special  melodic  figure,  and  two 
between  the  final  and  initial  chords  of  two  sections.  "In  this  position," 
he  says,  "their  employment  is  by  no  means  rare." 

Shinn  does  not  propose  to  sanction  consecutive  octaves  when  they 


XIV]  PROHIBITIONS  OF  CONSECUTIVES  117 

occur  in  connexion  with  discords.    Here  we  find  further  verification  of 
the  greater  power  of  the  octave  towards  bad  effect. 

As  regards  fifths,  Front's  rule^  may  be  taken  as  a  generally  accepted 
nucleus.  His  statement  that  the  rule  for  fifths  is  much  more  frequently 
broken  by  great  composers  than  the  rule  for  octaves  is  well  borne  out 
by  the  relative  frequency  of  examples  to  be  found  in  text-books  of 
harmony.  From  the  various  writers  I  have  consulted  (17,  23,  35,  38; 
45,  (48),  58,  61)  I  have  collected  some  fifteen  examples  of  octaves  and 
over  sixty  examples  of  fifths.  This  relation  may  seem  somewhat  strange 
in  view  of  the  fact  that  it  is  customary  to  speak  of  consecutive  octaves 
rather  lightly  and  as  being  objectionable  merely  because  of  the 
temporary  loss  of  distinction  between  the  two  voices^.  But  it  is  obvious 
that  this  theory  was  merely  a  deduction  from  the  notion  of  the  musical 
equivalence  of  octaves;  it  did  not  properly  reflect  the  nature  of  the 
musical  phenomenon  itself.  And  some  writers  even  proceed  to  explain 
the  bad  effect  of  fifths  by  the  loss  of  independence  of  the  voices  they 
appear  in. 

As  regards  the  influence  of  the  progression  referred  to  by  Prout, 
Gladstone  wrote  that  "of  the  various  exceptions  which  the  great 
composers  have  made  to  their  rule  of  avoiding  fifths,  none  are  more 
common  than  those  in  which  the  progression  is  either  from  the  tonic 
to  the  dominant,  from  the  tonic  to  the  subdominant,  or  the  reverse 
of  either"  (17,  I03f.).  This  is  the  first  stage  of  his  argument  in  favour 
of  explaining  the  effect  of  fifths  by  the  relative  position  of  the  chords 
in  which  they  occur.  It  is  again  to  be  regretted  that  his  statements 
were  not  accompanied  by  some  evidence  showing  relative  frequencies. 
For  the  next  degree  of  relationship  (a  third  between  the  roots)  he  could 
only  cite  two  cases  with  an  ascent  of  a  sixth  between  the  roots  and  none 
with  a  third. 

In  connexion  with  fifths  Shinn  (58,  280  fit.)  seems  to  rely  entirely  upon 
harmonic  relationship,  making  no  allusion  to  the  movements  of  the 
voices.   He  indicates  a  decreasing  frequency  of  occurrence  and  a  loss  of 

*  Of  contrary  motion  A.  Day  wrote:  "Fifths  by  contrary  motion  should  not  be  used 
(although  by  most  writers  allowed),  as  the  reason  given  why  fifths  by  similar  motion 
should  not  be  used  [they  give  the  idea  of  two  different  keys]  is  equally  applicable  to  fifths 
by  contrary  motion  "  ( 10,  lo).  So  much  the  worse  for  the  reason  given,  one  might  rather  say ! 

*  Pearsall,  for  instance,  dismisses  consecutive  octaves  and  unisons  in  five  lines  (of 
his  27  page  quarto  pamphlet),  saying  they  ought  to  be  avoided  because  of  'awkwardness,' 
and  "because  they  produce  no  effect  except  that  of  rendering  insipid  and  almost  nullifying 
any  harmony  of  which  they  may  be  component  parts"  (49,  m). 


118  EXCEPTIONS  TO  THE  [ch. 

effect  in  the  series  of  relationships  of  a  fifth  or  fourth,  a  third  or  sixth, 
and  a  second  or  seventh.  In  the  last  case  the  effect  is  rarely  satisfactory — 
except  for  special  purposes — in  the  root  position  of  the  chords;  it  is 
better  in  inversions.  Shinn  gives  many  examples  of  sequences  of  fifths 
in  connexion  with  discords^  (essential  and  unessential)  and  suspensions. 
The  explanation  he  offers  of  these  is  that  the  dissonant  note  imparts 
to  the  chords  "such  a  new,  distinctive,  and  relatively  speaking,  forcible 
character,  that  the  unpleasantness  due  to  the  consecutive  fourths 
[fifths],  if  it  is  not  obliterated,  is  neutralised  by  the  introduction  of 
this  new  element  and  the  sound  of  the  progressions  becomes  entirely 
satisfactory.  Not  only  is  this  explanation  perfectly  adequate,  but 
it  is,  we  believe,  the  only  one  which  it  is  possible  to  supply  that  is 
based  upon  the  musical  effect  of  such  progressions"  (58,  286 f.). 

This  last  exception  claimed  by  Gladstone  and  Shinn  is  confirmed  by 
the  large  number  of  instances  of  it  that  are  to  be  found  in  writers  on 
harmony.  I  have  collected  and  compared  more  than  sixty  cases  of 
consecutive  fifths,  mostly  from  the  greatest  and  most  accepted  com- 
posers^. Of  these  11  show  a  progression  from  a  discord  (commonly 
a  minor  seventh)  to  a  concord,  16  from  a  discord  to  a  discord,  and  15 
from  a  concord  to  a  discord.  In  reckoning  these  numbers  I  have  counted 
as  one  only  one  type  of  progression  in  a  given  work.  Thus  a  seventh 
to  a  seventh  would  reckon  as  one,  no  matter  how  often  it  were  repeated ; 
but  a  seventh  to  a  common  chord  ending  the  passage  would  count  as 
a  new  case.  Only  five  of  the  42  cases  are  by  contrary  motion.  Sixteen 
lie  between  the  bass  and  the  soprano  (B-S),  10  are  B-T,  8  are  S-A, 
7  are  A-T,  and  only  one  is  S-T.  In  resolving  these  sevenths  naturally 
pass  sometimes  to  dominant  and  to  tonic;  but  this  is  a  feature  of  the 
progression  that  must  be  considered  accidental ;  it  is  not  what  legitimates 
the  succession  of  fifths. 

That  can  only  be  the  discord  in  qu^tion.  And  the  effect  produced 
is  a  reasonable  one^.    In  this  type  of  exception  the  two  ends  of  the 

^  Day  held  that  fifths  by  contrary  motion  are  allowed  if  either  of  the  chords  or  both 
be  one  of  the  fundamental  sevenths  (10,  59). 

2  The  examples  collected  by  Parry  (48,  n9f.)  that  he  dubs  with  the  scornful  title 
"music-hall  cadence"  do  not  all  deserve  so  severe  condemnation  when  considered  in  this 
connexion,  whatever  other  faults  they  may  exemplify. 

^  Sacchi  said  consecutive  fifths  were  admitted  "(1)  When  the  fifths  lie  between  the 
inner  parts,  not  between  the  outer  ones,  (2)  when  the  fifth  with  the  bass  is  covered  by 
the  sixth,  (3)  when  the  two  fifths  are  so  placed  that  the  first  ends  one  period,  while  the 
second  forms  the  beginning  of  another."  Sacchi's  explanation  of  his  second  exception 
is  excellent:  "The  fifth  being  here  covered  by  the  sixth,  this  consonance  cannot  be  clearly 


XIV]  PROHIBITIONS  OF  CONSECUTIVES  119 

fusional  series  are  put  into  operation  against  one  another,  both  being 
intervals  characteristic  of  chords — not  a  sort  of  repeated  not«,  as  the 
octave  may  often  be.  The  consonance  makes  too  much  (perpendicular) 
unity  or  fusion,  the  dissonance  too  much  (perpendicular)  duality  or 
ruption  for  proper  melodic  flow.  But  their  combination  gives  a  new 
balance  of  flow.  Only, — and  that  may  well  be  a  notable  practical  point, 
— the  combination  is  most  frequently  such  an  interlocking  of  the  two 
elements  as  prevents  either  from  standing  forth  and  dominating  the 
progression. 

This  is  exemplified  in  my  sample.  In  16  cases  the  fifth  lay  B-S; 
of  these  only  two  are  by  contrary  motion.  The  beneficial  effect  of 
contrary  motion  is  evidently  not  required  for  B-S.  In  7  cases  the  position 
of  the  fifths  is  A-T,  and  all  of  these  are  by  similar  motion.  Ten  cases 
lie  between  the  bass  and  the  tenor  or  the  voice  next  above  the  bass. 
In  this  distribution  we  should  expect  the  fifths  to  be  more  liable  to  fall 
away  from  the  rest  of  the  chord,  and  so  to  become  more  than  usually 
noticeable.  This  view  is  perhaps  supported  by  the  fact  that  in  three 
of  these  cases  contrary  motion  has  been  used,  that  in  two  the  harmony 
is  of  six  parts,  that  one  is  produced  merely  by  a  sort  of  shake  in  the 
tenor,  and  that  in  two  one  of  the  voices  is  helped  to  continuity  by  an 
inserted  passing  note.  Thus  where  there  is  special  danger  of  the  inter- 
locking of  the  two  opposed  elements  in  the  chords  being  lost,  there 
the  composers  have  brought  other  compensatory  influences  to  bear 
upon  the  fifths.  In  one  case  S-A  with  similar  motion  the  progression 
is  to  a  chord  of  the  minor  seventh,  but  support  is  given  by  special 
melodic  features,  "the  carrying  out  of  a  thoroughly  established  idea," 
as  Parry  (45)  says  of  this  example.  In  one  by  Chopin  the  fifths  occur 
as  a  tremolo-like  accompaniment  to  the  bass  figure.  Five  others  are 
by  Dvofdk,  and  one  by  Stainer. 

Amongst  the  cases  involving  no  discords  the  fifths  are  referable  in 

enough  distinguished.  The  sixth  is  here  much  more  noticeable  than  the  fifth,  because 
it  is  the  extreme  parts  that  most  strike  the  ear  and  draw  the  greatest  attention.  Besides, 
the  two  neighbouring  notes,  the  fifth  and  the  sixth,  form  a  dissonance,  a  so-called 
acciacatura;  and  the  effect  of  the  acciacatura  is  a  certain  suspension  and  indecision  of 
sound,  that  makes  us  expect  its  resolution." 

The  admirable  Sacchi,  in  fact,  ends  on  an  ultra-modem  note  by  saying  that  we  must 
exercise  moderation  in  judging  of  consecutive  fifths,  and  that  wo  must  always  consider, 
beddofl,  "the  beauty  and  novelty  of  the  thoughts,  the  regularity  and  artistry  of  the 
progressions,  the  elegance  and  clearness  of  the  melod}',  the  unity  of  the  design,  the  force 
of  the  expression,"  and  the  'convenevolezza  del  costume'  (tht>  proprietj'  of  the  feeling?). 
In  short  we  have  to  be  equipped,  not  only  with  "the  eyes  of  the  fare,  but  with  those  of 
the  mind"  (57,  si  fl..  86  (.).   Plenty  of  scope  for  freedom  and  progress  there ! 


120 


EXCEPTIONS  TO  THE 


[CH. 


ten  cases  to  the  use  of  passing,  or  more  or  less  purely  ornamental,  notes. 
They  are  all  by  similar  motion. 

The  remainder  include  two  by  Mozart,  one  by  Mendelssohn,  one  by 
Rheinberger,  the  famous  one  from  Beethoven's  Pastoral  Symphony, 
three  by  Schumann  (one  between  the  beginning  and  end  of  phrases) 
and  one  by  Elert.  Four  of  the  preceding  are  by  contrary  motion. 
Two  (Prout  and  Gounod)  are  evidently  intended  to  give  the  effect  of 
barbarous  progression. 

Among  the  non-discordant  cases  that  by  Karg  Elert  (23,  lo)  is  not 
only  very  beautiful,  but  at  the  same  time  unique  in  its  build.  It  would 
seem  as  if  the  two  series  of  neutral  intervals  (sixths  and  tenths)  were 
able  to  outweigh  that  of  the  fifths  : 

Karg-Elert's  Example 
Karg -Elert,  "  Naher  mein  Gott  " 


Lento. 


lO- 


\ 


i 


f 


r:^: 


si: 


rJ         — 


:^= 


99 


:?=: 


1221 


Tzun^- 


rZJ I -^ 


±1 


irsL 


^ 


f 


999 


^-p- 


i^ 


Sw.  Salicional. 


While  progressions  (involving  fifths)  between  tonic  and  dominant 
or  subdominant  of  course  occur,  many  other  connexions  appear, in  the 
most  pleasing  cases.  From  the  cases  I  have  collected  I  cannot  persuade 
myself  that  the  harmonic  connexion  of  chords  (with  the  probable 
exception  of  the  one  just  mentioned)  plays  the  important  part  in  the 
degree  of  acceptability  of  consecutives  that  Gladstone  and  Shinn 
suggest.  But  I  do  not  wish  to  put  forward  the  groupings  I  have  just 
given  as  more  than  mere  suggestions.  The  subject  calls  for  an  extensive 
treatment  on  statistical  (and  experimental)  lines,  and  for  that  the  small 
number  of  examples  I  have  collected  is  only  profitable  in  broadest 
outline.  In  all  probability  they  form  a  very  special  and  biassed  example. 
But  there  is  no  doubt  that  the  effect  of  the  discords  on  the  fifths  stands 
out  prominently  even  so.  And  the  features  of  the  pleasing  uses  of  fifths 
thay  reveal,  generally  seem  to  be  compatible  with  the  theory  of  the 


XIV]  PROHIBITIONS  OF  CONSECUTIVES  121 

basis  of  the  prohibition  that  was  advanced  in  the  preceding  chapter. 
They  all  either  obliterate  the  special  effect  of  fifths  or  they  go  to 
strengthen  what  the  fifths  weaken — the  even  flow  of  the  different 
voices. 

For  consecutive  fourths  with  the  bass  it  is  stated  by  Shinn  (58, 166) 
that  the  prohibition  does  not  apply  to  progressions  in  which  either  of 
the  chords  forming  the  progression  is  a  discord.  Prout,  as  we  saw, 
allowed  them  only  "  when  the  second  of  the  two  is  a  part  of  a  fundamental 
discord  or  a  passing  note"  (cf.  58,  285 f.). 


CHAPTER  XV 

HIDDEN  OCTAVES  AND  FIFTHS,  ETC. 

In  all  treatises  on  harmony  a  regular  counterpart  to  consecutive  octaves 
and  fifths  is  found  in  the  special  treatment  of  so-called  hidden  octaves 
and  fifths.  The  traditional  explanation  of  the  latter  sets  them  into 
direct  connexion  with  the  former.  Two  voices  that  approach  a  consonance 
by  similar  motion  are  supposed  thereby  virtually  to  present  consecutives. 
Only,  as  the  tones  leading  to  the  consonance  are  not  really  sounded, 
they  were  styled  'hidden'  (cf.  55).  The  theory  is  ingenious  in  so  far 
as  it  thus  leaves  only  one  thing  to  be  explained, — the  bad  effect  of 
consecutives.  But  Shinn  (59)  is  not  far  wrong  in  saying  that  this  theory 
"can  only  be  regarded  as  an  interesting  tradition  of  the  past,  which, 
in  the  present  day  no  intelligent  musician  can  pretend  to  believe." 
We  are  the  more  relieved  from  discussing  it,  as  it  rests  not  on  any  facts, 
but  on  a  mere  assumption,  namely  that  the  listener  unconsciously  fills 
out  these  slurs.  Some  objective  strength  might  be  given  to  the  position 
by  claiming  that  there  is  some  sort  of  a  real  melodic  slur  involved  in 
the  process ;  but  even  that  theory  would  hardly  get  beyond  a  preliminary 
formulation.  For  this  melodic  slur  would  not  bring  the  phenomenon 
into  relation  with  consecutives  for  which  the  presence  or  activity  of  a 
melodic  slur  has  never  been  assumed. 

As  we  have  learned  in  the  preceding,  our  best  method  will  be  to 
find  and  to  describe  the  system  of  facts  of  which  '  hidden '  consecutives 
form  a  part  and  to  try  to  draw  from  this  system  an  interpretation 
sufficient  to  describe,  and  in  describing  to  explain,  its  members  and 
their  relations.  And  as  the  objective  facts  summarised  by  exponents 
of  systematic  harmony  themselves  suggest  the  procedure,  we  may 
take  our  systematic  table  of  consecutives  as  a  model  or  ideal  of  discovery 
in  dealing  with  '  hidden '  intervals.  As  before  we  shall  set  out  from  the 
formulations  of  E.  Prout  (52),  which  have  been  summarised  in  the 
previous  manner  in  Table  III. 

Table  III 

'  Exposed '  Intervals  (52  28  ff.) 

Showing  relations  between  ( I )  the  stringency  of  prohibition,  (2)  the  grade  of 
fusion  of  any  interval,  and  (3)  the  prominence  of  the  voice-parts  in  which  the  interval 
appears.    'Exposed'  intervals  (or  'hidden'  octaves,  fifths,  etc.)  are  such  as   are 


CH.  XV]         HIDDEN  OCTAVES  AND  FIFTHS,  ETC. 


123 


approached  by  'similar'  motion  in  the  two  parts,  i.e.  both  parts  rise  or  fall  (in 
pitch)  to  the  interval  in  question. 


0. 

5. 

4. 

Third  or 
sixth 

T 

Seventh 

Second 
or  ninth 

B-S 

forb. 

forb. 

+ 

+ 

? 

? 

1 
? 

B-A 

+ 

+ 

+ 

+ 

+ 

+ 

B-T 

+ 

+ 

+ 

+ 

+ 

+ 

S-A 

+ 

+ 

+ 

+ 

+ 

+ 

S-T 

+ 

+ 

+ 

+ 

+ 

+ 

A-T 

+ 

+ 

+ 

+ 

+ 

+ 

forb.  =  forbidden  with  exceptions;  f.  =  hardly  forbidden; 
+  =  allowed;  ?=  recommendation  against. 

The  formulations  embodied  in  the  Table  are  not  held  to  be  exact  laws,  as  it 
were,  but  only  to  be  properly  representative  of  the  trend  of  opinion  on  the  subject 
in  its  grading  of  stringency  from  point  to  point  of  the  Table. 

The  rules  upon  which  this  Table  is  based  are  the  following  (52,  28  fF.)  : 

(1)  Hidden  octaves  are  forbidden  between  the  extreme  parts;  except,  first, 
between  primary  chords  in  root- positions — (i.e.  with  the  roots  in  the  bass),  when 
the  bass  must  rise  a  fourth  or  fall  a  fifth,  and  the  upper  part  must  move  by  step; 
second,  when  the  second  of  the  two  chords  is  a  second  inversion',  the  bass  note  being 
either  the  tonic  or  dominant  of  the  key;  and  3rd,  when  the  second  chord  is  another 
position  of  the  first. 

Hidden  octaves  are,  however,  allowed  between  any  other  of  the  parts  than  the 
two  extreme  parts,  with  one  important  exception.  It  is  strictly  forbidden  to  move 
from  a  seventh  or  ninth  to  an  octave  by  similar  motion  between  any  two  parts, 
when  one  part  moves  a  second,  and  the  other  a  third.  This  is  the  very  worst  kind 
of  hidden  octaves,  and  must  be  most  carefully  avoided. 

(2)  Hidden  fifths  are  forbidden  between  extreme  parts;  except,  first,  in  a  pro- 
gression between  primary  chords  (tonic  to  dominant,  or  subdominant  to  tonic), 
with  the  upper  part,  as  with  hidden  octaves,  moving  by  step.  The  first  of  the  two 
chords  is  not  (as  in  the  case  of  octaves)  restricted  to  root-position  (two  examples 
are  given  of  it  in  first  inversion);  second,  from  the  root  position  of  the  chord  of 
the  supertonic,  with  the  third  in  the  upper  part,  to  the  chord  of  the  dominant, 
when  the  bass  falls  a  fifth,  and  the  upper  part  falls  a  third;  and  third,  from  one  to 
another  position  of  the  same  chord,  exactly  as  with  hidden  octaves. 


^  Here  there  is  sharp  opposition  with  Tchaikovsky  who  says  (76,  so):  "Concealed 
octaves  sound  particularly  unpleasant  when  the  octave  appears  as  doubled  Fifth  in  the 
six-four  chord  (or  as  doubled  third  in  the  sixth)."  In  both  examples  given  by  Prout 
as  good  the  octaves  are  given  by  a  doubled  fifth.  This  disagreement  between  the  two 
theorists  is  not  dne  to  confusion  of  the  requirements  of  the  hidden  octave  with  those  of 
the  fourth  from  the  bass,  but  reflects  their  attitude  towards  the  hidden  octave  only. 


124  HIDDEN  OCTAVES  AND  FIFTHS,  ETC.  [ch. 

(3)  When  two  notes  making  a  dissonance  with  one  another  (such  as  second, 
seventh,  or  ninth)  are  taken  without  preparation — that  is,  if  neither  of  them  has 
been  sounded  in  the  same  voice  in  the  preceding  chord — it  is  better  that  they  should 
enter  by  contrary  than  by  similar  motion,  especially  in  the  extreme  parts. 

Tchaikovsky  (76,  59£f.)  has  treated  hidden  octaves  and  fifths  in  some 
detail.  Of  the  examples  he  gives  as  bad  all  would  be  forbidden  by 
Prout,  except  the  progression  to  a  six-four  chord  with  doubled  fifth 
cited  above;  Tchaikovsky,  besides,  allows  the  octaves  or  fifths  "when 
they  occur  in  the  connexion  of  a  triad  with  the  Dominant  chord,  and 
when  the  seventh  is  prepared  in  an  inner  voice,"  a  case  that  Prout 
does  not  mention.  Tchaikovsky's  classifications  overlap  one  another 
considerably,  so  that  it  is  difficult  to  see  clearly  how  far  they  coincide 
logically  with  Front's.   They  may  be  summarised  as  follows  : 

(1)  Octaves  and  fifths  in  the  outer  voices  are  often  very  disagreeable 
and  should  be  avoided  by  all  means  :  (a)  when  the  upper  voice  proceeds 
by  a  jump,  but  even  then  the  progression  is  by  no  means  unpleasant 
in  the  connexion  of  a  triad  with  the  dominant  chord  when  the  seventh 
is  prepared  in  an  inner  voice;  (6)  when  all  the  voices  move  in  parallel 
motion;  (c)  when  the  outer  voices  move  in  parallel  jumps,  even  though 
the  inner  voices  remain  stationary  or  progress  stepwise. 

Here  there  is  evidently  an  opposition  between  jumps  and  steps  or 
stationariness  :  (a)  one  jump  in  the  upper  voice,  (b)  one  jump  worsened 
by  more  parallel  motion,  (c)  two  jumps  bettered  by  steps  or  stationariness. 
Prout  refers  to  step  only  in  the  upper  voice,  seeming  thus  to  indicate 
greater  importance  in  the  melodic  continuity  of  that  voice  (even  over 
the  bass)^. 

(2)  Concealed  progressions  between  an  outer  and  an  inner  voice 
are  disagreeable  when  the  jump  lies  in  an  outer  voice.  (C.  H.  Kitson — 
30,  50 — formulates  a  similar  rule  for  three  parts  as  against  four  parts, 
for  which  it  is  not  required.) 

(3)  Between  the  inner  voices  concealed  octaves  are  "entirely  out 
of  the  question,  arising  solely  in  consequence  of  bad  voice-leading"; 
"  concealed  fifths,  however,  are  permitted,  provided  the  voice  leading  is 
natural."  (The  word  'however'  seems  to  imply  that  the  octaves  are 
here  meant  to  be  more  forbidden  than  the  fifths.) 

On  the  whole  we  may  infer  a  grading  of  prohibitions,  greatest  for 
B-S,  least  for  A-T  and  medium  for  the  intervening  pairs  (cf.  55),  while 
the  minimum  for  the  fifth  in  A-T  is  less  than  for  the  octave.  Thus 
we  get  from  Tchaikovsky's  analysis  the  same  general  result  as  from 

1  But  cf.  below,  p.  130,  note  2. 


XV]  HIDDEN  OCTAVES  AND  FIFTHS,  ETC.  125 

Prout's.  Of  the  examples  given  by  Prout  as  good  or  bad  all  would  be 
similarly  styled  by  Tchaikovsky  except  (1)  the  six-four  case  already 
noted,  (2)  the  progression  to  another  position  of  the  same  chord,  which 
Tchaikovsky  neither  mentions  nor  exemplifies,  and  Prout's  second  excep- 
tion to  the  rule  against  hidden  fifths,  which  stands  in  systematic  isolation 
amongst  Prout's  other  rules,  as  far  as  one  can  see. 

Jadassohn  (26,  36,  70)  likewise  forbids  concealed  octaves  and  fifths 
when  both  voices  leap,  "no  matter  in  what  connexions  and  in  what 
direction,  if  in  outer  or  inner  voices,  or  one  outer  and  one  inner  voice," 
except  when  it  is  a  case  merely  of  inversion,  or  of  interchange  of  voices. 
One  leap  is  not  so  bad,  especially  if  it  is  not  in  the  upper  voice  (Sopr.), 
even  in  a  progression  between  the  triad  on  the  mediant  to  that  on  the 
submediant,  or  similarly  from  the  submediant  to  the  supertonic  (26,  37). 
There  is  some  sign  of  greater  freedom  with  the  fifth  than  with  the 
octave  :  for  "if  the  upper  voice  moves  by  a  degree,  and  one  of  the  lower 
voices  by  a  skip,  the  concealed  fifth  between  all  the  voices  is  without 
hesitation  permitted,  provided  all  the  voices  do  not  move  in  the  same 
direction^"  (26,  72);  in  the  case  of  octaves  a  slight  reservation  seems  to 
exist  in  the  case  of  connecting  the  chords  on  the  second  and  fifth 
degrees  (26,  37).  In  the  chord  of  the  dominant  seventh  a  fifth  may 
occur  even  after  similar  motion  of  all  the  voices,  but  not  along  with 
a  diminished  fifth. 

The  exceptional  treatment  of  the  progression  from  a  chord  on  the 
second  to  one  on  the  fifth  degree  may  have  some  connexion  with  Prout's 
second  exception  to  the  rule  for  hidden  fifths,  but  it  seems  to  contradict 
it  rather  than  to  support  it.  The  systematic  connexions  of  this  case 
are  not  at  all  evident.  Jadassohn  also  cites  the  modifying  effect  of  pitch- 
direction  on  this  succession:  "Concealed  octaves  in  the  succession  of 
the  chords  on  the  second  and  fifth  degrees,  when  downward,  are  not 
faulty,  as  this  connexion  does  not  sound  harsh"  (26,35);  they  must, 
however,  be  avoided  in  upward  progression  (26,  36).  These  points  do 
not  affect  the  main  result,  but  only  point  to  other  modifying  factors, 
which  would  each  require  a  separate  treatment,  i.e.  a  search  for  their 
systematic  setting. 

One  other  theorist  is  worthy  of  mention.   Shinn  (59)  says  : 

Viewed  from  the  modern  standpoint,  the  term  'hidden'  itself — seems  a  mis- 
nomer.... When  so-called  hidden  octaves  and  fifths  do  produce  an  objectionable 
musical  effect,  it  is  obviously  due  to  the  fact  that  the  octave  or  the  fifth  which 
is  present  is  it-self  thrown  into  undue  prominence,  and  what  is  unsatisfactory  in 

»  Cf.  below,  p.  126. 


126  HIDDEN  OCTAVES  AND  FIFTHS,  ETC.  [ch. 

the  progression  is  the  result  of  the  exceptional  'exposure'  of  the  perfect  interval 
which  is  present,  and  not  to  a  faulty  progression  which  is  supposed  to  exist  in  the 
imagination  of  the  listener.  This  fact  is  now  admitted  by  the  more  progressive 
amongst  musical  theorists,  and  in  modern  text-books  the  traditional  and  misleading 
term  'hidden'  is  being  gradually  displaced  by  the  more  accurate  one  'exposed,' 
the  universal  adoption  of  which  is  certainly  desirable  in  the  interests  of  all  students 
and  teachers  of  Harmony  (cf.  58,  265 a.). 

It  is  disappointing  to  find  so  interesting  a  statement  made  without 
any  citation  of  the  author  of  this  change  of  interpretation.  I  have  not 
as  yet  met  with  the  term  'exposed'  in  any  other  writer  on  harmony 
except  C.  H.  Kitson  (30,  49)  and  prefer  meanwhile  to  consider  F.  G.  Shinn 
himself  to  be  the  author  of  any  distinct  theory  of  'exposure'  in  these 
intervals,  as  he  is  certainly  the  only  satisfactory  exponent  of  the  idea. 
It  is  obvious  that  his  interpretation  is  much  more  in  line  with  the  trend 
of  our  own  exposition  than  is  the  older  term  'hidden,'  although  the 
latter  originated  quite  properly  in  an  attempt  to  connect  this  set  of 
facts  with  that  regarding  consecutives.  The  mistake  the  traditional 
theory  made  was  to  cling  to  the  mere  external  form  presented  by  the 
consecutives  as  the  essence  of  their  offensiveness  instead  of  getting 
behind  that  form  to  the  true  essence.  That  essence,  we  have  seen,  is 
most  probably  the  break  in  the  usual  melodic  connexion  due  to  the 
greatness  of  the  consonance  or  dissonance  in  question  and  to  its 
'exposure'  by  similar  motion. 

Shinn  then  points  out  that  "  some  of  the  most  eminent  continental 
theorists  impose  restrictions  upon  the  employment  of  octaves  and 
fifths  so  formed  [exposed],  either  between  two  inner  parts  or  between 
one  inner  and  one  outside  part,  which  are  not  recognised  by  English 
theorists."  He  then  cites  Tchaikovsky  and  Jadassohn  and  remarks 
in  connexion  with  an  example  constructed  to  fit  the  rule  quoted  from 
Jadassohn  above  (p.  125,  note)  : 

When  such  restrictions  and  prohibitions  extend  so  far  as  to  describe  as  bad 
and  to  forbid  the  employment  of  ["such  a  progression,"]  it  is  doubtful  whether 
such  rules  are  not  merely  devoid  of  all  musical  authority,  but  whether  they  possess 
any  value  even  for  the  purposes  of  mental  discipline;  whether,  in  fact,  they  do  not 
tend  to  make  the  introduction  of  any  kind  of  spontaneous  musical  thought  into 
the  work  of  the  student  absolutely  impossible. 

He  himself  then  proceeds  to  consider  their  employment  "from  the 
point  of  view  adopted  by  English  theorists,  that  is,  when  they  are 
formed  between  the  outside  parts";  and  he  suggests  on  the  basis  of 

the  examination  of  many  examples  of  exposed  octaves  and  exposed  fifths... that 
the  effect  of  such  progressions  varies  (and  is  more  or  less  satisfactory)  according 


XV]  HIDDEN  OCTAVES  AND  FIFTHS,  ETC.  127 

to  the  extent  to  which  the  particular  effect  of  the  exposed  interval  dominates  the 
effect  of  the  second  chord  of  the  progression.  When  the  nature  of  the  individual 
chords  forming  the  progression,  their  harmonic  relationship,  the  number  of  parts 
employed,  or  their  general  progression,  is  such  as  to  neutraUse  the  effect  of  the 
exposed  interval,  no  unsatisfactory  effect  is  produced. 

By  the  nature  of  the  individual  chords  is  meant  whether  either  or  both  of  the 
chords  be  a  discord,  or  whether  both  are  concords.  When  the  second  chord  is  a 
discord,  to  whatever  extent  an  exposed  interval  may  be  thrown  into  prominence, 
the  effect  of  this  exposure  is  almost  invariably  neutralised  by  the  dissonant  character 
of  the  chord. 

Shinn  gives  two  examples,  one  for  the  octave  and  the  other  for  the 
fifth,  each  occurring  in  a  chord  of  the  minor  seventh  (on  supertonic  and 
on  dominant). 

He  then  discusses  the  efEect  of  harmonic  connexion  upon  '  exposure '  : 

In  connexion  with  triads  and  their  inversions  whose  roots  are  a  fourth  or  a  fifth 
apart,  and  which  therefore  have  one  note  in  common,  when  one  of  the  parts  moves 
by  step,  the  other  part  leaping,  the  effect  is  rarely  unsatisfactory.  When  both 
parts  leap,  especially  in  a  downward  direction,  neither  part  leaping  more  than  a 
fifth,  the  effect  may  be  excellent. 

When  the  roots  of  chords  are  a  third  or  sixth  apart,  the  efEect  depends 
partly  upon  the  degrees  of  the  scale  upon  which  the  chords  stand, 
"some  being  bold  and  strong,  others  weak  and  unsatisfactory."  Down- 
ward approach  is  usually  a  favourable  circumstance.  The  degrees  used 
also  affect  triads  on  adjacent  notes.   Inversion  also  : 

When  one  of  the  chords  is  an  inverted  form,  and  the  highest  part  moves  by  step, 
while  the  lower  part  leaps  either  a  fourth  or  a  fifth,  the  effect  is  almost  invariably 
good....  An  exposed  octave  formed  between  two  such  triads,  the  second  being  in 
its  second  inversion,  is  also  unobjectionable  when  one  part  moves  by  step  and  the 
other  leaps  a  foiu^h....  The  strong  and  characteristic  movement  of  the  two  parts, 
by  a  second  and  by  a  fourth  or  fifth,  exerts  considerable  influence  in  the  direction 
of  strengthening  the  effect  of  the  progression  (cf.  68,  272  tr.). 

Here  we  see  a  number  of  factors  influencing  the  essential  factor  in 
question — the  degree  of  consonance  of  the  ('exposed')  interval;  we 
can  see  to  some  extent  the  direction  of  the  influence,  whether  it  is 
favourable  or  unfavourable  and  we  can  perhaps  assess  them  against 
one  another.  But  the  general  result  is  hardly  as  clear  as  is  desirable  for 
scientific  aesthetic  purposes.  Finally,  he  points  out  that  "progressions 
which  are  unsatisfactory  in  two  or  even  three  parts  may  be  quite  good 
in  four  or  five  parts."  This  he  says  is  due  partly  to  the  doubUng  of 
notes  common  to  both  chords,  partly  to  the  distribution  of  the  hstener's 


128  HIDDEN  OCTAVES  AND  FIFTHS,  ETC.  [ch. 

attention  over  a  greater  number,  when  "the  progression  of  any  two 
parts  (even  when  they  are  outside  parts)  must  in  some  corresponding 
proportion  become  less  noticeable"  (cf.  50,  255 f.,  296). 

The  result  of  this  analysis  of  representative  authors  may  seem 
hopelessly  confused,  as  it  must  certainly  be  highly  unsatisfactory  from 
any  practical  point  of  view^.  But  we  can  skim  the  common  general 
effect  off  the  differences  peculiar  to  the  different  writers.  (1)  The 
objectionable  phenomenon  in  question  is  presented  by  the  octave  and 
the  fifth  (all  theorists  agree),  and  somewhat  more  strongly  by  the 
octave  than  by  the  fifth  (Prout,  Tchaikovsky,  Jadassohn  and  Shinn)^. 
But  whereas  in  the  case  of  consecutives  the  objectionable  feature 
appeared  regularly  and  was  only  mitigated  or  made  tolerable  by  contrary 
motion,  in  this  case  (2)  it  does  not  appear  until  it  is  'exposed'  by  the 
unfavourable  effect  of  similar  motion.  Parallel  motion  in  other  voices 
than  those  primarily  concerned  increases  the  unfavourable  effect  (T.,  J.). 
(3)  The  effect  is  worst  in  the  outer  voices  (P.,  T.,  S.),  it  is  least  in  the 
inner  voices  (T.)  and  of  medium  degree  between  an  outer  and  an  inner 
voice  (T.). 

The  unconcern  of  the  English  exponents  to  any  but  the  outer  voices 
is,  of  course,  not  inconsistent  with  a  gradation  of  the  acceptable  effect 
in  the  rest  of  the  series  of  voice  pairs.  Differences  of  opinion  on  the 
position  of  the  point  of  change  from  desirable  to  undesirable  effect  are 
inevitable  in  the  midst  of  so  many  fluctuating  factors.  It  would  be  a 
mistake  to  wish  for  that  reason  to  throw  aside  all  the  formulations  of 
previous  exponents,  even  if  we  feel  that  their  terms  do  not  properly 
express  our  judgments,  and  to  start  a  search  for  an  entirely  new  set 
of  formulative  notions.  A  prejudice  of  that  kind  may  make  success 
impossible.  We  should  keep  our  attention  chiefly  directed  upon  the 
systematic  trend  of  the  rules  of  previous  analysts,  in  case  our  difference 
from  them  may  simply  be  settled  by  a  shifting  of  the  border  line  of 
acceptability  without  any  radical  change  of  the  basis  of  judgment. 

This  counsel  holds  not  only  for  the  primary  features  of  the  situation, 
but  also  for  the  minor  factors  which  create  exceptions  from  the  main 
rules. 

Thus  (4)  the  progression  by  step  has  a  favourable  effect,  while  leaps 

^  Kitson  says  (30,  *»):  "It  is  impossible  to  find  any  basis  of  general  agreement  as  to 
which  exposed  consecutives  are  objectionable  and  which  not." 

'  Shinn  says  "the  effect  of  exposed  octaves  distinctly  differs  from,  and  is  often  far 
less  satisfactory  than  that  of  exposed  fifths "  (58,  271). 


XV]  HIDDEN  OCTAVES  AND  FIFTHS,  ETC.  129 

are  nnfavourable.  Two  jumps — one  in  each  voice — are  worse  than  one, 
while  the  effect  of  any  one  jump  varies  with  the  prominence,  especially 
the  melodic  prominence,  of  the  voice  that  bears  it.  A  step  accordingly 
is  more  powerful  in  the  highest  voice^,  then  in  the  lowest,  etc.  Jadassohn 
says  jumps  in  any  two  voices  are  bad,  while  one  is  least  bad  if  it  is  not 
in  the  upper  voice.  Tchaikovsky  deems  two  jumps  in  the  outer  voices 
very  bad,  one  jump  worse  in  an  outer  than  in  an  inner  voice,  and  not 
usually  admissible  in  the  highest  voice.  Prout  requires  a  step  in  the 
upper  voice  to  support  the  favourable  influence  of  close  relationship 
of  chords.  And  Shinn  thinks  the  strongest  effect  is  produced  when  the 
upper  part  moves  by  step  (58,  272ff.)2.  He  also  desires  two  jumps  (in 
outer  voices,  of  course),  to  be  mitigated  by  downward  motion  and 
restriction  of  the  leap  to  the  'emmelic'  range  (not  more  than  a  fifth). 
Melodic  continuity,  then,  counteracts  the  bad  effect  of  the  consonance 
exposed  by  similar  motion.  This  agrees  with  our  conclusion  for  con- 
secutives,  that  their  offence  was  a  breach  of  the  required  melodic 
continuity.  (Two  steps  would,  of  course,  often  convert  the  problem 
into  one  of  consecutives.  A  tritone  may  be  followed  by  a  perfect  fifth 
if  the  lower  part  rises  a  semitone;  then  there  are  two  'steps.'  Cf.  p.  104 
above,  Rule  II.)  But  while  there  is  thus  agreement  in  the  general 
trend,  there  is  considerable  disagreement  between  writers  as  to  the 
margin  of  pleasant  effect.  But  the  drawing  of  these  border  lines  does 
not  primarily  concern  us  here. 

(5)  Writers  (T.,  J.,  S.)  agree  that  the  bad  effect  is  covered  over 
again  in  characteristic  discords,  e.g.  that  of  the  minor  seventh.  One 
might  argue  that  the  exposure  should  then  be  greater  as  the  two  intervals 
stand  at  the  ends  of  the  fusional  scale.  But  the  direction  of  pull,  as  it 
were,  is  opposite  in  the  two  cases  :  one  is  consonance  or  unity,  the 
other  is  dissonance  or  duality.  Perhaps  this  combination  achieves 
something  like  the  neutrality  of  the  thirds  and  sixths  (without,  of 
course,  rendering  the  chord  consonant  as  a  whole). 

(6)  'Connexions'  between  the  chords  or  the  tones  seem  favourable. 
When  the  chords  only  differ  in  position,  the  bad  effect  is  annulled, 
probably    because   melodic   connexion   is   readily   attained  from   one 

*  Which  in  modem  music  is  usually  not  only  generally,  but  also  thematically,  melodic. 
Cf.  Tovey's  definition  of  melody  as  "the  surface  of  music." 

*  In  the  case  of  the  tritone  followed  by  the  fifth,  already  referred  to  (p.  1 04),  the  progression 
is  satisfactory  in  so  far  as  the  leading  tone,  which  forms  the  lower  note  of  the  tritone, 
moves  by  semitone  to  the  tonic.  But  if  this  movement  is  exposed  on  the  bass,  it  so  strongly 
'announces'  the  coming  fifth  that  we  get  a  specially  exposed  fifth,  or  else  the  leading  tone 
refuses  its  chief  function  by  fsdling  (cf.  62,  37,  loi). 

w.  F.  M.  9 


130  HIDDEN  OCTAVES  AND  FIFTHS,  ETC.  [ch. 

position  to  another  of  the  same  chord  or  pattern.  Here  the  second 
tonal  mass  has  been  facihtated  for  the  mind  by  the  occurrence  of  the 
first  one,  so  similar  to  it.  Thus  the  melodic  connexions  are  favoured. 
Similarly  when  the  bass  moves  from  one  important,  and  therefore 
familiar  (or  easy),  point  of  the  tonal  system  to  another  (tonic  to  dominant 
and  vice  versa,  subdominant  to  tonic  and  vice  versa),  or  when  it  merely 
moves  by  similar  intervals,  fourth  and  fifth),  continuity  in  the  bass 
is  made  easy;  and  if  step  relieves  any  difficulty  in  the  upper  voice,  the 
effect  should  be  excellent '^.  But  there  seems  no  strong  reason  why 
with  Shinn  these  relations  should  not  be  inverted  to  step  in  bass  and 
easy  leap  (e.g.  fourth  or  fifth  in  upper  voice).  Only,  the  system  of  grada- 
tions we  have  shown  would  incline  us  to  expect  not  quite  so  good  an 
effect  in  the  latter  way  as  in  the  former^.  But,  of  course,  both  ways 
may  be  acceptable.  The  degrees  of  the  scale  used,  other  than  tonic, 
dominant,  and  subdominant,  may  further  modify  the  ease  of  progression 
according  to  their  familiarity  in  the  tonal  system. 

(7)  Front's  note  regarding  the  seventh  or  ninth  before  the  octave 
seems  clear,  because  these  intervals,  even  in  isolation,  strongly  suggest 
the  unity  and  repose  of  the  octave.  This  suggestion  is  probably  even 
increased  by  the  stepwise  movement  of  one  of  the  voices,  which  will 
ordinarily  be  accompanied  by  the  movement  of  a  third  in  the  other 
voice,  when  the  following  interval  is  an  octave.  Thus  the  octave  about 
to  be  heatd  will  be  strongly  suggested,  and  so  will  be  more  exposed 
than  ever. 

(8)  In  dealing  with  consecutives  we  have  already  encountered  the 
effect  of  a  larger  number  of  parts  than  four.  Prout  (52,  305)  says  of 
this  :  "  In  proportion  as  the  number  of  parts,  and  therefore  the  difficulty, 
increases,  the  stringency  of  the  rules  relaxes.    These  hidden  fifths  and 

^  These  points  are  nearly  all  applicable  to  the  unison,  which  may  not  generally  be 
taken  by  similar  motion,  unless  in  the  progression  from  dominant  to  tonic,  with  the  help 
of  step  in  one  voice  also  perhaps  (cf.  52,  sif.).  The  voices  in  which  the  unison  appears 
will  generally  be  neighbouring  ones,  of  course.  The  unison  by  similar  motion  seems  to 
be  more  strictly  forbidden  than  is  even  the  octave.  The  unison,  of  course,  gives  the 
clearest  expression  to  the  characteristic  of  consonances — their  apparent  unitariness. 

*  This  would  imply  that  the  leap  of  a  fourth  or  fifth  is  a  more  potent  factor  than  is 
a  step.  For,  as  the  bass  gives  greater  exposure  than  the  soprano,  x  (fourtih  or  fifth  in  bass) 
+y  (step  in  soprano)  would  be  greater  than  y  (step  in  bass)  +  a;.  This  inference  might 
be  preferable  to  that  suggested  on  page  124  above  (greater  melodic  prominence  of  the 
soprano),  which  is  irreconcilable  with  the  general  predominance  of  the  bass  except  in 
so  far  as  the  soprano  in  modern  music  is  often  thematically  the  most  melodic  voice,  though 
it  can  never  be  generally — or  in  its  mere  sonorousness — the  most  melodic.  Cf.  the  later 
discussion  on  the  most  general  aspect  of  melody. 


XV]  HIDDEN  OCTAVES  AND  FIFTHS,  ETC.  131 

octaves  are  allowed,  even  when  both  voices  leap;  consecutive  octaves 
and  fifths  by  contrary  motion  may  be  used  freely;  we  even  meet  in  the 
works  of  the  great  masters  with  examples  of  a  doubled  leading  note, 
though  it  is  better  to  avoid  this,  if  possible."  It  would  probably  be 
wrong  to  suppose  that  the  mere  difficulty  of  the  work  is  the  basis  of 
license,  as  if  an  ugliness  could  be  compensated  by  such  an  extraneous 
reason.  It  seems  preferable  to  suppose  that  the  succession  is  admitted 
because  the  many  parts  steady  each  other  and  maintain  a  general 
distribution  of  effects.  The  obscurity  of  the  inner  parts  in  four  part 
writing  has  the  same  origin. 

We  shall  have  some  opportunity  later  of  considering  the  influence 
of  inversions.  And  the  difference  between  upward  and  downward 
motion  may  be  neglected  for  the  present;  it  is  not  quite  clear  what  the 
relation  of  this  difference  is  to  the  matters  referred  to  in  chapter  viii. 
The  implication  is  that  descent  strengthens  melodic  continuity,  possibly 
because  descent  has  more  of  the  character  of  a  return  to  the  starting- 
point,  while  ascent  is  departure  (cf.  above,  p.  52  f.).  Front's  unique 
rule  regarding  the  supertonic  chord  is  probably  more  or  less  of  an 
accident  in  his  usually  so  systematic  work.  It  seems  a  composite  of 
downward  motion  and  of  progression  of  a  fifth  in  bass.  Only  the  latter 
factor  is  invoked  by  Shinn  to  explain  the  case  (58,  49). 

In  conclusion  we  may  say  that  the  Table  at  the  beginning  of  this 
chapter,  based  upon  Front's  formulations,  properly  represents  the 
trend  of  differences  it  refers  to,  and  may  therefore  be  taken  as  valid 
materia]  upon  which  a  theory  of  the  phenomenon  in  question  may  be 
raised.  As  we  have  so  often  suggested,  the  exact  place  at  which  one 
draws  the  line  between  what  is  desirable  and  what  not,  depends  on 
variable  subjective  factors.  But  the  scale  of  differences  which  make 
different  aesthetic  reactions  possible  is  objective  and  invariable,  as  is 
also  the  general  trend  of  their  influence  upon  the  aesthetic  reactions 
of  an  individual.  The  aesthetic  realm  is  not  to  be  considered  the  sport 
of  caprice  because  there  is  no  disputing  about  tastes.  The  latter  state- 
ment is  true  only  because  one  man  may  carry  subjective  inclinations 
and  influences  about  with  him  that  another  does  not  possess  and  finds 
of  no  particular  interest.  If  these  so  affect  him  as  to  make  his  aesthetic 
judgments  contrary  to  another's,  the  latter  cannot  offer  to  dispute  him 
out  of  their  influence.  Between  one  man  and  another  only  those  forces 
are  subject  to  common  analysis  or  discussion  that  are  objective  to 
both  of  them,  or  that  are  primarily  rooted  in  the  artistic  work  itself. 

»— 2 


132  HIDDEN  OCTAVES  AND  FIFTHS,  ETC.  [ch.  xv 

A  study  of  the  factors  that  operate  in  some  individuals  and  not  in 
others  is  work  for  a  psychology  of  personal  differences,  not  for  a  science 
of  the  constitution  and  potencies  of  the  aesthetic  objects  themselves. 
About  these  and  the  laws  of  their  being  dispute  is  as  little  excluded  as 
it  is  in  dealing  with  nature  itself. 

It  is  often  said  that  artistic  rules  are  hardly  formed  before  they  are 
swept  away  into  obUvion  by  the  stroke  of  some  genius.  That  and  all 
such  expressions  are  radically  wrong,  as  wrong  as  it  would  be  to  say 
that  no  sooner  is  a  law  of  nature  discovered  than  some  engineer  sweeps 
it  away  by  showing  how  to  circumvent  it  or  (apparently)  to  oppose 
and  to  reverse  its  action.  The  will  of  a  person  may  be  overruled  and 
forgotten,  but  no  one  supposes  the  art  of  a  past  century  to  have  been 
a  tyrant's  will,  A  genius  breaks  no  rule  of  art.  He  only  fulfils  it  the 
more  by  finding  influences  which  unite  with  it  to  produce  effects  it 
would  be  incapable  of  producing  alone.  After  all  no  one  really  believes 
this  fable  of  the  genius.  You  always  have  to  be  the  genius  before  you 
can  have  his  power  to  make  rules  disappear.  You  must  have  his  know- 
ledge and  experience.  In  fact,  you  must  know^  how  to  do  it.  And  it 
is  right  that  the  beginner  should  learn  the  big  facts  and  rules  first,  as 
really  and  permanently  valid,  not  irreconcilably  valid  or  as  pedants' 
foibles  to  be  discarded  later.  He  must  first  learn  the  broad  effects  and 
then  progress  towards  the  subtle  ones;  and  he  will  do  so  easily  and 
wilUngly  when  his  discipline  can  be  set  before  him  in  systematic  form. 
When  its  foundations  have  been  well  expounded,  he  will  be  able  of 
himself  to  carry  them  forward  into  their  interactions  with  one  another 
much  more  readily  and  steadily  than  he  would  if  he  had  to  learn  them 
all  separately  and  unintelligently. 

^  Or  'feel'  (in  sensory  constructions).  A  composer  who  does  so,  may  perhaps  not  ako 
know  the  method  or  law  inherent  in  his  feeling.  It  is  for  the  theorist  to  find  that  out,  if 
the  composer  does  not  first  discover  it  himself.  In  his  "  Philosophy  of  Modernism  (in 
its  connexion  with  music)"  Cyril  Scott  says:  "It  is  a  fact,  almost  a  truism,  among 
enlightened  musicians,  that  we  leam  the  rules  only  in  order  to  know  how  to  break  them ; 
but  the  real  quarrel  arises  in  how  often  and  how  far  one  is  permitted  to  break  them. 
The  truth  is,  in  reality  there  are  no  rules.  There  are  merely  conventions;  and  these 
conventions  have  altered  with  the  advent  of  each  new  master"  (p.  27 f.).  'Breaking 
them '  suggests  the  overcoming  of  law  by  its  better  fulfilment;  but,  of  course,  there  is 
also  such  a  thing  as  the  mere  ignoring  of  rules. 


CHAPTER  XVI 

A  FOURTH  FROM  THE  BASS 

It  i8  a  notable  fact,  which  was  emphasised  in  the  preceding  chapter, 
that  similar  motion  is  the  primary  condition  of  the  prohibited  'exposure ' 
of  the  two  intervals  octave  and  fifth.  The  next  most  important  condition, 
embodied  in  the  Table  on  page  123,  is  the  prominence  or  exposure  of 
the  pair  of  voices  that  bear  the  interval.  We  noticed  that  amongst 
English  theorists  the  prohibition  of  exposure  extends  only  to  the  most 
prominent  pair — bass  and  soprano.  The  interval?  prohibited  are 
familiarly  octave  and  fifth  only ;  but  it  is  proper  to  place  the  dissonances 
at  the  other  extreme  of  the  same  Table  because  of  the  common  desire 
that  they  should  not  be  exposed  by  similar  motion. 

We  have  thus  detected  at  least  a  part  of  a  system  of  facts  that 
forms  a  proper  counterpart  to  the  system  obtained  for  consecutives. 
We  might  offer  to  pass  the  rest  of  the  system  as  the  counterpart  of 
successive  thirds  and  sixths — the  neutral  region  for  which  no  prohibition 
exists.  The  only  difference  would  be  that  this  neutral  region  is  now 
wider.  And  that  would  be  in  natural  agreement  with  the  greatly 
reduced  degree  of  prohibition  set  upon  'exposed'  intervals.  We  might 
even  infer  from  the  scope  of  this  widening  that  the  fourth  as  a  consonance 
borders  more  closely  upon  the  neutral  region  than  its  common  ranking 
as  a  perfect  consonance  would  suggest.  That  would  also  agree  with 
the  system  of  consecutives;  for  in  it  we  saw  that  only  fourths  from  the 
bass  are  forbidden.  Or  we  might  say  that  the  fourth  lies  on  the  border 
between  high  grade  consonance  and  neutral  sonance. 

But  we  cannot  accept  any  such  simple  solution  without  careful 
inquiry.  For  it  is  a  familiar  fact  of  harmony  that  a  fourth  from  the 
bass  must  be  used  with  great  care.  Many  rules  for  its  use  have  been 
formulated  and  these  must  be  carefully  analysed  before  we  can  judge 
accurately  of  the  status  of  the  fourth  as  an  '  exposible '  interval. 

We  shall  again  base  our  summary  in  the  first  instance  upon  the 
formulations  of  E.  Prout  (52,  66fi.). 

"Though  it  is  possible,"  Prout  says,  "to  take  any  triad  in  its  second 
inversion,  the  employment  of  any  but  primary  triads  in  this  position 
is  extremely  rare."    Macfarren  (35,  68  ff.)  had  asserted  that  only  those 


134  A  FOURTH  FROM  THE  BASS  [ch. 

on  the  tonic,  subdominant,  and  dominant  were  admitted.  It  is,  of 
course,  the  '  root '  of  the  chord  that  is  supposed  to  stand  on  these  degrees ; 
the  lowest  tone  of  the  second  inversion  is  a  fourth  lower.  For  purposes 
of  description  these  two  authorities  may  be  said  to  agree.  The  six- 
four  chord  has  not  only  to  be  approached,  but  also  to  be  quitted,  in 
certain  ways. 

It  may  be  approached  :  (1)  either  by  leap  or  by  step  from  the  root 
position  of  another  chord;  (2)  by  leap  from  another  position  of  the 
same  chord;  (3)  by  step  (but  not  by  leap)  from  the  inversion  of  another 
chord;  (4)  from  a  different  chord  upon  the  same  bass  note.  These 
steps  and  leaps  all  refer  to  the  bass  note  upon  which  the  fourth  stands. 

It  may  be  left :  (1)  by  step  of  a  tone  or  a  semitone,  upwards  or  down- 
wards, and  the  following  chord  may  be  either  in  root  position  or  in  an 
inversion;  (2)  by  the  same  bass  note  or  its  octave  bearing  another 
chord,  provided  the  six-four  chord  is  a  tonic  or  a  subdominant  chord. 
(This  is  the  'cadential  six-four.')  And  in  this  case  the  six-four  must 
be  on  a  strong  accent,  unless  it  has  also  been  preceded  by  a  chord  on 
the  same  bass  note.  (Evidently  this  cadential  effect  requires  a  sort  of 
rhythmical  exposure  of  the  six-four  chord  to  create  a  ready  disposition 
for  it  in  the  hstener's  mind.)  (3)  By  leap  to  another  note  of  the  same 
chord  (without  change  of  harmony),  provided  that,  when  the  harmony 
changes,  it  returns  either  to  its  former  note,  or  to  the  note  next  above 
it  or  below  it.   In  other  cases  it  is  not  good  for  it  to  leap  (32,  70f.). 

Now  we  know  already  that  for  melodic  continuity  a  leap  is  less 
favourable  than  a  step,  and  the  repetition  of  the  same  note  is  easier 
to  follow  than  is  a  step.  By  abstracting  this  term  from  the  above 
rules  and  by  grading  those  that  are  left  over  against  it,  we  may  attempt 
to  gauge  the  effect  of  the  latter.  By  this  process  Table  IV  has  been 
constructed.  In  order  to  enter  a  single  value  at  each  point  I  adopted 
as  indicator  the  extreme  permitted  under  each  head.  Thus  'by  leap  or 
by  step'  means  'even  by  leap';  if  a  leap  is  permitted,  so  is  a  step, 
of  course.  Similarly  'inversion'  means  'either  inversion  or  a  more 
favourable  form — root  position.'  But  'root  position'  means  'only root, 
not  inversion.' 

The  Table,  then,  indicates  that  the  melodic  continuity  and  ease  of 
the  bass  voice  in  a  six- four  chord  are  a  matter  of  special  concern.  For 
when  favourable  conditions  are  secured  for  it  by  conjunct  motion  or 
by  no  change  at  all,  the  other  parts  have  complete  freedom  of  movement ; 
but  when  its  movement  is  made  less  cogent  by  a  leap,  the  others  should 
be   more  favourable,   giving  at  least  either  the  same  chord   or   (in 


xvi] 


A  FOURTH  FROM  THE  BASS 


135 


approaching  the  six-four)  proceeding  from  the  clear  cut  stability  of  a 
root  position. 

Table  IV 

Scheme  of  rules  {after  E.  Provt,  52,  70f .)  for  approaching  and  quitting  a 

six-four  chord. 


Approaching  a  ^ 

Quitting  a  | 

Movement  of 
the  Bass 

Prom—  /  k\ 
chord      V  I  / 

In-       /*\ 
position  \     J 

chord  \      / 

In— 

position 

By  leap 

By  step 

None  (or  octave) 

Same        )  i 
Different  \ 

Different 

Different 

]  Inversion 
4  1  root 

Inversion 

Inversion 

Same  (1) 

Different  (3) 
Different  (2) 

Inversion 

Inversion 
Inversion 

The  arrow-heads  point  in  the  direction  of  greater  melodic  continuity  and  ease.  Each 
entry  in  the  Table  indicates  the  maximum  allowed. 

Note  1.  Provided  that  when  the  harmony  does  change,  it  returns  either  to  its  former 
note  or  to  the  note  next  above  it  or  below  it  (or  to  a  note  to  which  a  correct  progression 
from  the  six-four  chord  could  have  been  made,  58,  6i). 

Note  2.  Provided  the  six-four  chord  is  a  tonic  or  a  dominant  chord.  And  in  this 
case  the  six-four  must  be  on  a  strong  accent,  unless  it  has  also  been  preceded  by  a  chord 
on  the  same  bass  note  (cadential  six-four). 

Note  3.  In  this  case  the  six-four  should  occur  on  the  unaccented  part  of  the  measure 
(76,  Mf.).  If  the  one  is  followed  by  another  six-four,  the  latter  will  be  cadential,  and  so 
a  rule  of  frequency  for  successive  six-fours  may  be  formulated  (cf.  38,  oo). 

The  rules  given  by  Prout  thus  evidently  form  a  consistent  whole, 
which  is  at  the  same  time  thoroughly  representative  of  the  generalisations 
of  other  writers  on  harmony.  Minds  may,  of  course,  as  we  have  already 
noted,  vary  for  subjective  reasons  in  the  exact  margin  of  desirability 
they  draw.  For  a  study  of  the  effect  of  objective  differences  that  concerns 
us  now,  the  essential  consideration  is  the  trend  of  the  changes  in 
desirability.  The  smaller  the  range  of  personal  differences,  the  more 
reliable  will  this  trend  be.  And  here  the  range  seems  to  be  very  small 
indeed. 


There  is,  then,  in  the  six-four  chord  a  sort  of  opposition  or  rivalry 
between  the  two  parts  of  the  chord, — the  bass  and  the  other  parts. 
The  problem  is  :  what  is  the  basis  and  origin  of  this  melodic  rivalry? 

A  strong  indication  is  given  by  the  fact  that  the  essential  charac- 
teristic of  the  chord  is  its  fourth  from  the  bass.  This  has  been  condensed 


136  A  FOURTH  FROM  THE  BASS  [ch. 

into  the  radical  classification  of  the  fourth  as  a  dissonance.  Such  a 
theory  is,  of  course,  erroneously  extreme  :  the  fourth  cannot  be  set 
down  generically  as  a  dissonance,  for  apart  from  the  bass  it  is  not  at 
all  dissonant.  But  the  indication  at  least  requires  us  to  consider  whether 
any  special  treatment  has  to  be  given  to  the  upper  note  of  the  fourth. 

Prout  makes  no  statement  on  this  point  whatever,  beyond  pointing  out 
that  the  upper  is  the  dissonant  note  and  therefore  should  not  be  doubled 
(52,  71).  Macfarren  (35,  so)  pointed  out  that  "the  consonance  of  the 
fourth  in  these  three  inversions  [on  tonic,  subdominant,  and  dominant], 
is  proved  by  the  entirely  free  progression  of  the  4th  it  comprises,  which 
is  the  assumed  dissonant  note  of  the  disputants."  Shinn  (58,  61)  says  the 
rules  for  six-fours  "refer  entirely  to  the  progression  of  the  bass  part." 
On  the  contrary  Tchaikovsky  says  that  the  six-four  chord  "must  in 
any  [other?]  case  be  connected  with  one  of  the  neighbouring  chords 
by  means  of  its  fourth;  and  into  and  from  its  other  neighbour  the 
fourth  must  progress  stepwise"  (76,56).  Mansfield  (38,38)  is  more 
comprehensive :  "The  4th  ...  being  a  dissonant  interval  should  be 
approached  and  quitted  conjunctly,  and,  if  possible,  in  contrary  motion 
with  the  bass.  Faihng  this,  it  should  be  prepared,  i.e.  heard  as  a  con- 
sonant note  (or  as  an  essential  note,  i.e.  a  note  without  which  a  chord 
would  be  incomplete,  such  as  a  7th  in  the  chords  of  the  dominant  and 
diminished  7ths)  in  the  same  part  in  the  previous  chord.  If  approached 
by  skip,  it  should  be  in  contrary  motion  with  the  bass,  and  it  should 
be  quitted  by  oblique  motion  with  the  bass  when  contrary  motion  is 
not  possible.  These  rules  apply  to  almost  all  dissonant  notes."  Table  IV 
shows  that  oblique  movement  is  frequently  implied  in  the  rules,  e.g.  over 
against  'leap'  and  'none.'  Opportunities  for  similar  motion  would 
arise  mainly  where  the  bass  moved  by  step.  Whether  the  number  of 
times  it  actually  occurs  is  so  small  as  to  make  a  prohibitory  rule  useful 
I  do  not  know.  One  of  Shinn's  examples  (58,  60,  e)  shows  similar  motion. 

In  favour  of  the  consonance  of  the  fourth  there  are  to  be  urged — 
the  work  above  expounded  regarding  fusion,  a  large  part  of  all  musical 
experiences  relating  to  the  fourth  (i.e.  all  apart  from  the  bass),  its 
graded  position  in  the  Table  of  consecutives,  and  the  beneficial  effect 
of  (fundamental)  discords  upon  consecutive  fourths  from  the  bass. 
The  beneficial  effect  of  contrary  motion  claimed  by  Mansfield  would 
also  correspond  to  its  ranking  as  an  'exposed'  interval  next  to  the 
fifth.  There  is  no  doubt  that  the  fourth  is  the  consonance  of  third  grade 
and  that  the  fourth  from  the  bass  is — to  some  extent  at  least — an  exposed 
interval. 


XVI]  A  FOURTH  FROM  THE  BASS  137 

On  the  other  hand  the  rules  for  its  use  seem  to  be  far  more  numerous 
and  stringent  than  we  might  have  expected  from  its  grading  after  the 
fifth.  Contrary  motion  does  not  seem  to  be  enough  to  reduce  the 
'exposure'  without  the  help  of  conjunct  motion  in  the  lower  voice 
or  even  in  both.  And  if  consecutive  fourths  from  the  bass  are  condoned 
in  discords,  we  should  expect  to  find  a  single  fourth  from  the  bass 
a  fortiori  tolerable  in  a  discord.  But  Prout,  for  example,  says  (52, 104) 
that  "the  rules  for  approaching  and  quitting  a  second  inversion  apply 
to  the  second  inversions  of  discords  as  well  as  of  concords." 

This  last  statement  seems  to  dispose,  not  only  of  the  bass  fourth 
as  an  '  exposed '  interval,  but  as  a  dissonance  as  well.  Why  in  approaching 
and  leaving  an  interval  that  is  merely  supposed  to  be  a  dissonance 
should  we  have  to  take  so  much  more  care  than  in  dealing  with  intervals 
that  are  undoubtedly  dissonant?  If  the  rules  are  too  complex  to  allow 
us  to  assume  the  consonance  of  the  bass  fourth,  they  are  just  as  excessive 
for  its  dissonance.  The  characteristic  of  dissonances  that  appears  in 
the  fourth  is  its  tendency  to  suggest  the  major  (or  minor)  third;  it  seems 
to  call  for  a  resolution  into  that  interval  and  to  urge  the  whole  chord 
in  which  it  appears  in  that  direction.  But,  in  contrast  to  regular 
dissonance,  we  find  that  this  characteristic  can  be  suppressed  by 
appropriate  circumstances.  The  discordant  feature  can  be  eliminated. 
In  no  regular  dissonance  do  we  find  that  any  method  of  approach  or 
departure  from  the  interval  will  make  it  appear  consonant,  however 
it  may  alter  the  trend  of  its  impulse  to  resolution,  or  facilitate  its 
appearance  in  the  music. 

A  trend  of  resolution  appears  most  strongly  in  the  'cadential  six- 
four  chord,'  which  is  subjected  to  a  rhythmical  'exposure.'  The  interval 
is  not  thereby  rendered  dissonant;  it  stands  forth  clearly  as  a  fourth; 
we  may  even  suppose  that  its  grading  as  a  consonance  emphasises  this 
exposure  to  some  extent;  we  have  no  reason  to  argue  that  in  that  case 
it  should  be  approached  by  contrary  motion,  which  would  reduce  the 
'exposure';  for  its  exposure  is  just  what  we  desire  at  the  moment, 
seeing  that  we  also  give  it  a  rhythmical  exposure.  But  there  are  only 
two  degrees  of  the  scale  that  properly  invite  this  cadential  resolution — 
the  dominant  and  the  tonic  (tonic  and  subdominant  chords).  These 
are  the  only  degrees  of  the  diatonic  scale  that  have  a  semitone  imme- 
diately below  them.  The  one  gives  a  cadence  on  to  the  dominant, 
the  other  on  to  the  tonic.  A  cadence  upon  the  subdominant  is  spoilt 
by  the  tritone  that  there  appears  instead  of  the  fourth.  Where  tonality 
is  well  marked,  the  dissonantal  tendency  of  the  fourth  is  most  useful. 


138  A  FOUKTH  FROM  THE  BASS  [ch. 

But  as  it  is  not  desirable  to  pursue  a  cadence  of  this  kind  whenever 
a  six-four  chord  is  used,  means  have  to  be  found  to  avoid  the  effect. 
This  is  done  by  placing  the  chord  upon  an  unaccented  part  of  the 
measure  and  by  leaving  it  by  step  in  the  bass.  The  other  procedure, 
noted  in  Table  IV,  note  1,  is  not  really  a  variant  upon  these  two.  It 
represents  only  a  temporary  movement  from  the  six-four  bass. 

The  bass  note  of  the  fourth  is,  therefore,  the  most  important  of  the 
chord,  in  so  far  as  progression  from  it  is  concerned.  If  it  is  exposed 
(1)  by  being  the  bass  note,  (2)  by  being  rhythmically  accented,  (3)  by 
being  the  tonic  or  dominant  degree  of  the  scale,  it  will  produce  the  effect 
of  arrest  of  motion  and  strongly  suggest  the  interval  of  the  major 
third  that  is  so  near  to  it,  and  so  produce  the  cadence.  If  such  a  cadence 
is  not  desired,  the  exposing  conditions  (3)  may,  and  (2)  must,  be  avoided, 
and  the  remaining  tendency  to  revive  the  cadential  tendency  must  be 
suppressed  by  giving  the  bass  a  strong  melodic  force — progression  by 
step. 

It  is  by  reference  to  this  tendency  towards  the  third  that  we  must 
explain  the  special  precautions  to  be  taken  in  approaching  the  six-four 
chord.  Even  when  the  bass  fourth  is  skilfully  introduced,  it  still  tends 
more  or  less  strongly  to  suggest  the  third.  All  the  more,  then,  should 
we  expect  to  find  a  tendency  to  confusion  of  melodic  attachments 
inherent  in  the  approach  to  a  bass  fourth.  Unless  special  care  is  taken 
the  melodic  streams  would  tend  to  fall  towards  the  third  and  a  jar 
of  surprise  would  be  caused  by  the  fourth  actually  given.  If  this  jar 
is  to  be  avoided,  we  must  lead  a  strong  melodic  current  upon  the  tones 
of  the  bass  fomth  and  especially  upon  the  lower  or  bass  tone  which 
gives  the  pitch  or  centre  to  the  whole  chord. 

Thus  we  see  that  the  problem  of  the  fourth  is  akin  to  that  of 
consecutives.  The  difficulty  in  both  is  the  maintenance  of  clear,  un- 
ambiguous melodic  lines.  But  they  are  otherwise  different.  In  con- 
secutives and  exposed  intervals  the  obstruction  is  chiefly  caused  by 
the  pronounced  consonance  or  dissonance  inherent  in  the  single  interval 
itself.  In  the  fourth  it  is  due  to  the  proximity  of  the  fourth  to  the 
interval  of  the  third.  The  distracting  influence  is  external.  How  it 
comes  to  have  this  effect  we  shall  consider  later  on. 

These  difficulties  and  uncertainties  into  which  we  have  been  led, 
show  us  more  than  ever  how  desirable  it  is  that  information  should 
be  gathered  about  the  treatment,  not  only  of  consecutive  intervals, 
but  also  of  all  single  intervals  in  relation  to  the  circumstances  under 


XVI]  A  FOURTH  FROM  THE  BASS  139 

which  they  are  used,  on  a  statistical  basis.  To  many  the  statistical 
method  may  seem  to  be  a  dry  as  dust  business.  But  if  we  are  to  over- 
come the  divergence  of  opinion  which  characterises  the  exposition  of 
harmony  and  to  obtain  a  body  of  definite,  generally  accepted  knowledge, 
it  is  our  only  hope.  No  other  method,  either,  will  ever  permit  us  to 
subtract  from  the  treatment  accorded  to  an  interval  the  part  that  is 
probably  due  to  a  certain  influence  so  as  to  leave  us  with  the  amount 
due  to  any  other.  These  apportionings  call  for  a  quantitative  treatment, 
and  that  is  procurable  only  by  statistical  methods.  At  the  present 
time  every  writer  who  works  out  the  rules  of  harmony  for  himself 
has  to  do  over  again  work  done  by  many  others  before  him.  And  he 
does  not  relieve  his  successors  of  the  necessity  of  repeating  the  work. 
At  best  he  can  cover  only  a  small  range  of  the  task,  and  in  doing  so  he 
is  liable  to  be  greatly  influenced  not  only  by  the  generalisations  of  his 
predecessors  but  also  by  his  own  special  preferences  and  prejudices. 
With  statistical  methods,  however,  a  piece  of  work,  if  it  is  once  done 
thoroughly,  is  not  only  finished,  but  is  open  to  the  view  of  every  one 
else.  It  need  be  repeated  at  most  only  once  for  the  purpose  of  verifica- 
tion. Anyone  who  desires  to  continue  research  of  a  problem  already 
investigated,  will  have  an  opportunity  in  testing  the  validity  of  statistics 
already  derived  from  one  composer  or  period  with  those  to  be  derived 
from  another.  By  this  means  a  reliable  history  of  the  developments  of 
harmony  would  in  time  result.  Once  the  methods  of  this  kind  of 
statistical  research  were  well  known,  such  repetitive  tasks  might  be 
given  to  younger  students  who  desire  to  follow  the  work  of  any  composer 
with  close  analytic  attention.  As  things  are  at  present  such  a  historical 
view  is  only  present  in  feeble  outline  of  the  broadest  kind.  We  do  not 
know  even  whether  the  formulations  of  the  best  analysts  are  complete 
or  how  far  they  still  fall  short  of  approximate  completeness. 

The  results  we  have  obtained  thus  far  alone  sufl&ce  to  convince  us 
that  with  the  proper  systematic  approach  and  outlook  there  is  every 
prospect  that  the  science  of  harmony  will  one  day  attain  a  high  grade 
of  precision,  if  statistical  methods  are  carefully  pursued.  It  will  become 
capable  of  systematic  treatment  that  should  make  its  apprehension 
easy  and  comprehensive.  This  possibility  seems  much  more  probable 
for  music  than  for  the  pictorial  arts.  Music  has  the  advantage  of 
operating  with  units  of  sound  that  are  capable  of  only  slight  fluctuations 
from  certain  forms — their  pitches.  There  are  perhaps  many  people 
who  would  look  upon  such  an  accurate  science  of  divinae  musicae  as 
a  disastrous  calamity.    But  that  is  really  an  absurd  point  of  view.   An 


140  A  FOURTH  FROM  THE  BASS  [ch.  xvi 

art  can  only  be  furthered  by  a  greater  knowledge  of  its  essential  nature. 
Its  progress  should  then  be  more  rapid  and  sure.  We  could  estimate 
the  possible  lines  of  advance  with  great  probability  of  success,  and  if 
no  new  vistas  seemed  likely  to  open  up  along  our  present  lines  of  progress, 
those  who  are  in  search  of  new  lands  would  know  to  what  point  of 
the  system  of  sounds  that  leads  to  music  they  would  have  to  recede 
in  order  to  be  able  to  diverge  upon  strange  paths  of  new  outlook.  And 
that  would  be  a  great  gain.  We  do  not  know  whether  much  of  the 
experimentation  in  music  of  to-day  is  not  from  the  outset  a  waste  of 
time.  With  a  science  of  music  well  developed  we  should  be  able  to 
judge  on  this  matter  with  some  certainty.  The  world  is  not  the  greater 
or  freer  to  a  genius  for  his  ignorance.  He  does  not  create  by  personal 
decree,  but  by  discovery  of  new  effects  which  were  already  laid  down 
as  possibilities  in  the  systematic  growth  of  the  art  before  he  took  it 
over.  And  all  unaided  discovery  is  slow  and  painful,  even  to  a  genius. 
With  knowledge  discovery  may  become  possible  to  many  others  besides 
the  genius,  who  may  then  climb  nearer  to  his  summits. 

In  bur  knowledge  of  the  physical  basis  of  pitch  we  have  a  very 
accurate  science  of  the  fundament  of  music.  Here  our  knowledge  is 
practically  complete.  But  no  one  supposes  that  the  divine  art  has 
become  any  more  earthly  for  that  reason.  Why  should  a  science  of  the 
art  itself  degrade  it  any  the  more?  Those  in  whom  knowledge  and  its 
precision  tend  to  dispel  the  attractions  of  beautiful  and  wonderful 
things^  will  still  be  able  to  keep  their  minds  unsullied,  if  they  so  desire. 
But  the  charm  of  mystery  does  not  lie  in  any  vagueness  of  the  sensory 
stuff  of  art  or  of  its  beauty,  but  in  all  the  longing  hopes  these  finished 
forms  arouse  in  our  minds.  We  feel  the  course  of  life  as  it  might  be, 
were  we  not  our  own  poor  guides  stumbling  towards  ends  we  can  only 
dimly  discern,  but  the  stuff  upon  which  some  divine  artist  had  chosen 
to  lay  his  wondrous  hand.  We  move  in  the  divine  thought  wrapt  up 
in  that  stuff  of  sound  and  we  long  to  have  and  to  be  its  life. 

Nevertheless  the  sounds  we  hear  have  the  precise  structure  of 
crystal  and  their  beauty  is  a  chiselled  gem.  Their  sciences  may  be  their 
equal  and  counterpart. 

1  Cf.  A.  E.  Hull,  Cyril  Scott,  London  1918,  p.  78  f.:  "  Like  Debussy,  he  [Cyril  Scott] 
would  protest  against  the  dissection  of  his  music,  as  if  it  were  a  piece  of  curious  clock- 
work mechanism.  In  the  Retme  Blatiche  in  1891  the  French  master  wrote,  "As  children 
we  were  taught  to  regard  the  dismemberment  of  our  playthings  and  toys  as  a  crime  of 
high  treason,  but  these  older  children  still  persist  in  poking  their  noses  where  they  are 
not  wanted,  endeavouring  to  explain  and  dissect  everything  in  a  cold-blooded  waj',  thus 
putting  an  end  to  all  mystery." 


CHAPTER  XVII 

COMMON  CHORDS  OR  CONCORDANCE 

From  chapter  xi  till  now  we  have  been  engaged  essentially  in  the  study 
of  single  intervals  of  two  tones.  It  is  true  we  have  considered  them 
generally  as  they  stand  in  harmony  of  four  or  more  parts.  But  our 
interest  centred  primarily  in  the  interval  of  two  tones  itself,  as  if  it 
were  the  element  of  structure  of  four  part  harmony.  The  results  of 
our  study  enable  us  now  to  show  the  exact  manner  in  which  intervals 
generally  follow  one  another  in  harmony  of  two  or  more  parts. 

These  results  seem  both  to  enrich  and  to  modify  the  outlook  afforded 
by  such  previous  knowledge  as  had  been  systematically  sifted.  That 
culminated  in  the  notion  of  a  fusion  inherent  in  each  pair  of  tones 
themselves  and  not  borrowed  from  any  of  their  adjuncts.  Consonance 
and  dissonance  were  the  opposite  poles  of  this  fusion.  And  consonance 
seemed  obviously  to  be  the  ground  upon  which  the  pleasures  of  music 
mainly  stand,  although  they  were  evidently  greatly  enhanced  by  contrast 
with  dissonance.  A  general  statement  of  this  kind,  however,  seemed 
plainly  unable  to  give  any  sort  of  adequate  expression  to  the  whole 
nature  of  music  in  many  parts.  The  theory  showed  a  crudity  and 
insuflSciency  very  hke  that  of  the  primitive  music  of  the  discantors 
in  comparison  with  the  modern  art. 

The  outlook  presented  by  the  concept  of  fusion  was  clouded  by  the 
emphasis  laid  upon  the  approximation  of  the  high  grade  fusions  to  the 
unity  and  balance  of  a  single  pure  tone,  and  the  ensuing  tendency  to 
carry  that  notion  over  into  the  general  idea  of  consonance  and  dissonance 
as  they  enter  into  modern  music.  Thus  the  function  expected  from 
an  octave  or  a  fifth  in  music  was  such  as  would  express  its  unity  or 
approximation  to  the  balance  of  a  single  tone.  The  function  of  a  second 
or  a  seventh  would  reveal  its  tonal  duaUty.  But  the  merest  glance 
at  the  nature  of  music  seemed  to  contradict  any  such  conclusion.  For 
in  the  prototype  of  musical  groupings  of  tones — in  the  common  chords — 
we  find  three  essential  tones  and  three  intervals.  Approximation  to 
the  balance  of  a  single  tone  is  out  of  the  question.  Even  a  dull  ear 
would  detect  plurality  in  every  case.  And  the  interval  of  the  chord 
most  essential  to  its  musical  functions  is  not,  as  the  theory  of  fusion 
would  most  likely  suggest,  its  fifth,  but  its  lower  third.   But  it  was  not 


U2       COMMON  CHORDS  OR  CONCORDANCE      [ch. 

apparent  from  the  previous  theory  why  the  lower  third  should  be  more 
important  than  the  upper  one.  In  fact  attempts  were  made  to  explain 
the  differences  of  major  and  minor  triads  in  terms  of  the  reversal  of 
the  positions  which  the  thirds  occupy  in  them  (43;  of.  60,  35 f.,  44-53 
(ZarUno),  219  ff.  (Rameau),  293  ff.  (Tartini),  367  ff.  (Hauptmann),  385  ff. 
(Oettingen),  387ff.  (Riemann)).  But  without  avail  (cf.  67,  84ff.;  71,333; 
79).  We  do  not  look  upon  the  one  chord  from  below,  and  upon  the  other 
from  above. 

The  great  importance  of  the  thirds  in  modern  music  did  not  seem 
to  fit  into  the  theory  of  fusion.  For  they  were  neither  high  grade 
consonances  nor  high  grade  dissonances,  A  means  of  reaching  their 
musical  function  seemed  indeed  to  ensue  upon  the  distinction  of  grades 
of  pleasantness  in  intervals.  Thirds  and  sixths  rank  high  in  the  scale 
(29, 194).  But  it  must  be  evident  that  mere  pleasantness  without  a 
justifiable  basis  of  pleasure  is  a  weak  reed  for  any  theory  of  music  to 
lean  upon  (cf.  71,  351  ft.). 

So  the  outlook  upon  music  seemed  to  be  blocked  completely.  There 
seemed  to  be  no  means  of  approach  to  music  as  we  find  it.  And  it  was 
inevitable  that  in  time  an  attempt  should  be  made  to  make  a  new 
start,  to  find  a  new  notion  upon  which  the  functions  of  chords  might 
be  grounded. 

This  idea  Stumpf  attempted  to  supply  in  his  notion  of  concordance 
as  distinguished  from  consonance  (71). 

Two  notes  are  consonant  when  they  sound  together  so  as  to  fuse 
into  an  approximate  unity,  whether  the  component  tones  are  distin- 
guished or  recognised  at  the  same  time  or  not.  The  greatest  degree  of 
unification  appears  in  the  octave.  It  lessens  progressively  in  the  fifth, 
fourth,  etc.,  while  in  the  dissonances  we  find  least  of  it.  Consonance 
and  dissonance  appertain  in  this  original  and  limited  sense  only  to 
every  two  tones.  "  Only  as  thus  understood,  as  the  relation  of  two  tones 
to  one  another,  is  consonance  the  basal  phenomenon  of  all  music" 
(71,329).  "It  must  always  be  borne  in  mind  that  what  I  call  fusion 
can  only  then  be  perceptible  as  such  when  the  fusing  tones  are  distinguished 
from  one  another;  just  as  we  cannot  perceive  similarities  without  keeping 
the  similars  separate.  But  if  this  is  done,  if  the  three  tones  of  a  trichord 
are  distinguished  from  one  another,  I  at  least  can  form  a  judgment  on 
their  fusion  only  by  pair- wise  comparison,  but  I  cannot  besides  discover 
a  fusion  that  attaches  to  the  whole,  to  the  triad  as  such"  (71,  330). 

It  must  be  evident  that  thus  far  at  least  the  empirical  teachings 


XVII]  COMMON  CHORDS  OR  CONCORDANCE  143 

of  harmony  gathered  together  in  the  previous  pages  and  the  results 
that  have  emerged  from  them  confirm  this  general  attitude  of  Stumpf's 
towards  intervals  quite  unambiguously.  Every  pair  of  tones  is  in 
harmony  a  distinct  individual,  as  it  were;  it  in  no  way  ceases  to  be 
itself  or  changes  into  another,  owing  to  the  simultaneous  presence  of 
other  tones.  As  that  individual,  it  carries  its  own  degree  of  'fusion' 
unchangingly  about  with  it,  although, — and  this  must  be  emphasised, — 
the  effect  produced  by  that  fusion  at  any  moment  is  to  some  extent 
modifiable  by  a  number  of  circumstances  other  than  the  fusion  itself. 
Moreover  the  appreciation  of  all  harmonic  effects,  even  of  such  elementary 
ones  as  we  have  as  yet  been  able  to  study,  presupposes  always  in  every 
musical  ear  some  sort  of  ability  to  distinguish  every  pair  of  voices  from 
every  other. 

What  kind  of  distinction  is  implied  is  left  unsaid.  Doubtless  it  may 
vary  greatly  in  degree  of  clearness.  Low  grades  of  distinction,  as  masical 
analysis  of  finer  order  would  rat«  them,  are  apparently  quite  good 
enough;  for  every  beginner  is  supposed  to  be  able  to  appreciate  readily 
enough  what  is  taught.  The  ease  and  certainty  of  analysis  that  is 
habitual  in  the  most  finely  endowed  musical  minds  is  by  no  means 
essential.  The  beginner  is  not  required  to  be  able  to  name  every  ordinary 
chord  as  soon  as  heard,  or  to  sing  its  components,  or  even  to  hear  them 
by  mental  analysis  singly,  or  to  separate  in  turn  each  pair  from  the 
others  in  his  mind's  ear.  It  is  enough  if  he  can  hear  and  attend  well 
enough  to  get  the  chief  effects  that  are  produced  by  any  pair  of  voices 
amongst  others  :  e.g.  the  bad  effect  of  consecutives,  exposed  intervals, 
etc.  That,  the  teachings  of  harmony  show  us  implicitly,  is  already 
hearing  the  tones  of  chords  pair  by  pair^. 

Another  point  stressed  by  Stumpf  is  that  "consonance  is  not  changed 
either  by  the  addition  of  a  third  or  fourth  tone.  What  is  changed  is 
the  musical  meaning  of  the  tones  and  their  pleasantness.  But  the 
unitariness  of  the  octave,  the  duality  of  the  seventh  survives  in  any 
and  every  arrangement  with  other  tones"  (71,  328).  That,  again,  is 
true,  but  only  in  the  sense  of  the  preceding  paragraph.  The  original 
essence  or  being  of  the  consonance  or  dissonance  is  not  altered,  but  the 
effect  of  it  or  its  suitability  or  its  functions  as  a  unit  of  musical  structure 
are  certainly  changed.  The  terms  used  by  Stumpf — the  musical  meaning 
of  the  tones  and  their  pleasantness — do  not  specify  what  these  functions 

*  Cf.  72,  B1-S7,  which  still  fails  to  bridge  the  gulf  between  the  usual  static  analysis 
of  tone-massea  and  the  fluid  analysis  of  musio. 


144  COMMON  CHORDS  OR  CONCORDANCE  [ch. 

are.    They  imply,  however,  that  the  degree  of  fusion  is  not  their  basis 
or  source. 

The  complicated  psychical  processes  that  bring  certain  modes  of  apprehension 
to  bear  upon  sensations  that  have  been  changed  either  only  subliminally  or  not  at 
all  (often  even  in  a  contrary  sense)  must  not  be  confused  with  the  simple  facts  of 
sense  perception  by  which  the  basal  phenomenon  of  all,  even  of  non-harmonic, 
music  is  given.  That  one  and  the  same  unmodified  pair  of  tones  should  now  fuse 
more  and  now  less  according  as  we  apprehend  it  as  c — el>  or  as  c — djf  is  out  of  the 
question,  because  fusion  is  a  function  of  the  two  sensations — or  of  their  physiological 
bases — and  can  change  only  with  these  same  (71,  aas). 

In  view  of  the  needs  and  practices  of  music  Stumpf's  attitude  towards 
his  notion  of  fusion  is  readily  intelligible.  A  generalised  notion  of  fusion 
in  the  sense  of  degree  of  unitariness,  applicable  to  any  tonal  mass  of 
however  many  components,  would  fail  to  solve  the  problems  of  musical 
science.  Stumpf  does  well  to  look  about  for  some  new  fundamental 
notion  that  will  meet  the  situation.  Nevertheless  it  remains  true  that 
the  notion  of  unitariness  is  logically  quite  as  applicable  to  any  number 
of  simultaneous  tones  as  to  two.  A  triad  cannot  but  approximate  more 
or  less  to  the  unitariness  of  a  single  tone,  even  if  we  add  the  proviso  : 
whether  its  component  tones  are  distinguished  or  not.  If  that  approxi- 
mation and  that  proviso  pass  for  two  tones,  they  must  be  equally  valid 
for  three  or  more.  Stumpf's  attempt  to  dam  up  the  logical  vitality  of 
the  concept  of  fusion  is  certainly  not  the  method  that  will  lead  us 
quickly  forwards.  The  procedure  makes  a  semblance  of  success  only 
so  long  as  the  waters  fail  to  overflow. 

But  let  us  notice  the  alternative  foundations  offered  : 

Our  music  rests  without  doubt  upon  the  trichord  in  its  two  forms  major  and 
minor.  The  question  then  is:  what  is  the  objective  justification,  the  reasonable 
principle  of  structure,  of  trichords?  This  question  is  usually  either  not  asked  at 
all  (as  in  the  most  of  the  text-books  of  harmony)  or  it  is  absolved  by  a  reference  to 
the  series  of  partials.  In  this  series  4:5:6  are  indeed  found,  and  further  on  the 
minor  trichord  10  :  12  :  15  as  well;  but  there  are  in  it  many  other  trichords  besides, 
that  are  not  honoured  in  such  a  way  by  music,  although  they  partly  have  even 
smaller  ratios  than  the  minor,  such  as  7:9:  11.  What  then  gives  these  two  chords 
their  dominating  position,  and  why  must  just  three  tones  be  bound  together  gene- 
rally, if  more  than  one  are  to  be  combined  at  all? 

The  fundamental  principle  may  be  formulated  thus  :  Let  the  greatest  number 
of  tones  within  the  octave  be  taken  that  are  severally  consonant  with  one  another, 
and  so  that  we  pass  in  the  tonal  motion  from  below  upwards  and  amongst  the 
consonances  from  the  stronger  to  the  weaker  degrees  of  consonance. 

Starting  from  any  tone  we  get  according  to  this  principle  first  its  upper  fifth, 
— so,  from  c,  g,  and  then  only  either  et>  or  e  is  further  possible,  if  we  neglect  for 
the  present  the  'sevens'  [5 :  7  and  such  like].    Thus  with  the  upper  finish  of  the 


XVII]  COMMON  CHORDS  OR  CONCORDANCE  145 

octave  there  result  the  two  chords  ce\fgc^  and  cegc^.  In  them  all  higher  grades  of 
fusion  are  represented.  But  as  it  is  at  once  apparent  that  c^  has  again  an  octave 
above  itself  and  within  this  new  octave-space  the  same  process  repeats  itself,  there- 
fore we  do  not  reckon  c^  further  as  a  part  of  the  structure  won  from  c,  but  as  funda- 
mental tone  of  the  analogous  one  an  octave  higher.  Thus  we  reach  the  trichord, 
in  its  two  forms  simultaneously  (71,  ssif.). 

Stumpf  then  proceeds  to  show  in  a  very  summary  way  how  the 
usual  chordal  combinations  of  modern  music  might  be  developed. 
Finally  he  gathers  the  results  together  in  special  concepts. 

As  a  chord  we  designate  a  group  of  simultaneous  tones... that  can  be  reduced 
in  the  way  indicated  to  chief  or  accessory  triads  of  a  certain  fundamental  tone. 
Tone-groups,  therefore,  with  dissonant  intervals  are  called  chords  in  this  sense, 
but  not  all  and  sundry,  only  those  that  can  be  obtained  from  triads  by  certain, 
operations  (71,  337). 

Chords,  therefore,  fall  into  two  classes.  Concords  (as  in  our  usual 
sense  of  the  word)  must  contain  a  fifth  or  its  inversion  a  fourth,  and  a 
third  or  a  sixth.  Discords  are  all  other  chords  in  the  sense  just  expounded. 
Concordance  and  discordance  are  the  corresponding  abstract  terms. 

Consonance  and  dissonance  are  thus  presupposed  by  the  notions 
of  concordance  and  discordance.  But  the  latter  notions  differ  from  the 
former,  which  apply  only  to  pairs  of  tones  and  to  tone  groups  only  in 
80  far  as  they  consist  of  pairs  of  tones.  Concordance  applies  primarily 
only  to  groups  of  three  or  more  tones  and  can  be  transferred  to  a  tone 
pair  only  if  and  in  so  far  as  it  is  apprehended  as  a  part  of  a  concord, 
i.e.  of  a  triad  (71,  340).  Thus  one  and  the  same  tone  pair  may  be  at 
one  moment  concordant,  at  another  discordant  according  to  the  setting 
in  which  it  is  apprehended.  It  is  the  setting  that  makes  the  difference. 
So  concordance  and  discordance  only  appear  with  at  least  three  tones. 
And  "consonance  is  a  matter  of  direct  sensory  perception,  whereas 
concordance  is  a  matter  of  apprehension  and  relational  thinking  "  (71, 34i). 

Stumpf  points  out  finally  that  these  expressions  are  not  by  any 
means  new  to  musical  theory.  "  But  since  Franco  [the  words  concordare, 
discordare  have  been  used]  perhaps  from  the  feehng  that  it  is  no  longer 
a  matter  of  merely  'sounding  together  or  sounding  apart,'  but  also  of 
'fitting  together  and  not  fitting  together'"  (71,  350). 

Now,  however  valuable  this  exposition  of  the  notion  of  concordance 
may  be  in  so  far  as  it  gives  an  account  of  the  character  peculiar  to 
groups  of  at  least  three  tones  or  to  intervals  as  parts  of  these  (compare 
the  notion  of  'pattern'  expounded  in  chapter  x  above),  certain  points 
call  for  immediate  remark. 

W.  F.  M.  10 


146  COMMON  CHORDS  OR  CONCORDANCE  [ch. 

(1)  Upon  what  real  ground  of  tonal  functions  does  the  alleged 
constitutive  principle  of  chords  rest?  None  has  been  given  or  even 
indicated.  A  logical  ground  alone  is  evident.  That,  of  course,  is  in 
itself  a  very  important  matter,  but  it  is  quite  powerless  to  make  between 
tone  pairs  and  triads  the  real  separation  that  Stumpf  claims.  It  could 
at  most  make  a  merely  logical  division,  such  as  would  divide  the  discus- 
sion or  study  of  tone  pairs  from  that  of  triads.  It  could  not  justify 
the  rule  that  consonance  applies  only  to  tone  pairs,  concordance  only 
to  triads  or  larger  groups  of  tones.  Nor  could  it  do  anything  to  constitute 
the  relational  thinking  that  is  claimed  as  the  essence  of  concordance. 
It  remains  as  great  a  mystery  as  ever  how  triads  with  their  three  tone 
pairs  come  to  form  the  basis  of  modern  art. 

(2)  No  doubt  Stumpf  is  firmly  convinced  that  consonance  is  always 
a  function  of  two  tones  at  a  time.  And  he  may  well  be  right  in  this. 
But  even  then  he  is  so  only  through  'knowledge  by  acquaintance,' 
not  through  'knowledge  by  description.'  In  other  words  he  feels  it  or 
knows  it  by  experience,  but  he  does  not  know  it  logically  or  scientifically. 
It  has  not  been  proved  by  him.  On  the  contrary  the  principle  upon 
which  concordance  is  founded  would  lead  us  to  expect  that  a  chord  is 
only  a  group  of  fusional  pairs  or  their  derivatives.  Then  there  would 
be  no  real  division  between  groups  of  two,  and  groups  of  three  or  more 
tones.  And  there  is  also  no  clear  reason  why  we  should  not  turn  our 
relational  thought  upon  a  succession  of  tone  pairs  as  well  as  upon  a 
sequence  of  chords  or  look  upon  concordant  triads  as  parts  of  discord- 
ances in  four  or  more  voices.  There  seems  to  be  no  such  radical  distinction 
between  music  of  two  parts  and  music  of  more  than  two  parts  as  Stumpf 's 
distinction  between  consonance  and  concordance  would  lead  us  to  sup- 
pose. 

(3)  If  two  tones  necessarily  make  some  approximation  to  the  unity 
of  a  single  tone  of  whatever  degree,  we  have  still — as  far  at  least  as 
Stumpf's  science  can  show — every  reason  to  expect  that  any  group  of 
tones  should  do  the  same.  Or  rather  we  should  expect  that  every  triad 
should  be  rather  more  dissonant  than  otherwise.  For  it  would  certainly 
not  tempt  us  to  take  it  for  a  unity,  whether  we  distinguished  its  com- 
ponent tones  or  not. 

(4)  Here  we  come  upon  an  important  point.  Stumpf's  dissonance 
is   more   or  less   a   negative   idea^,   like   Helmholtz's   consonance   in 

^  As  it  is  also  in  many  other  writers,  amongst  the  ancient  Greeks  for  example,  and 
in  Gevaert  himself  (cf.  above,  p.  108).  But  not  all  the  Greek  writers  neglected  the  positive 
aspect  of  dissonance  (v.  p.  154,  below). 


xvn]  COMMON  CHORDS  OR  CONCORDANCE  147 

simultaneous  intervals.  It  is  merely  a  minimal  degree  of  approximation 
to  unity,  of  unitariness  (with  or  without  distinction  of  the  component 
tones  or  pitches).  Is  that  enougM  Affirmation  would  imply  that  two 
minimally  unifying  tones  are  as  such  impleasant.  But  why  so?  There 
is  no  obvious  reason.  On  the  other  hand,  if  while  highly  unifying 
pairs  give  approximation  to  balance,  minimally  unifying  pairs  give, 
not  merely  a  non-unity,  but  a  positive  unbalance  or  irregular  confusion, 
we  should  be  able  to  bring  that  chaos — as  a  positive  ground  of  unpleasant- 
ness— into  connexion  with  similar  grounds  in  other  spheres,  e.g.  pictorial 
art,  logical  thought,  feeling,  etc. 

But  that  is  not  the  end  of  the  subject.  Evidence  has  been  brought 
above  to  show  that  there  are  grades  of  fusion  that  must  be  called 
neutral — neither  distinct  consonance  nor  distinct  dissonance.  Here  we 
come  upon  an  aspect  that  does  not  seem  to  be  subsumable  under  the 
fundamental  notion  of  fusion  as  approximate  unitariness.  Nor  can  we 
well  conceive  of  an  indifference-point  between  balance  as  approximation 
to  the  unity  and  symmetry  of  a  single  tone  and  unbalance  or  chaos. 
How  far  away  from  balance  should  we  have  to  fix  this  point?  But 
one  might  say  :  consider  the  middle  point  to  be  balance  and  suppose  a 
departure  from  it  in  two  directions,  one  towards  loss  of  balance  in  unity, 
the  other  towards  loss  of  balance  in  conflict.  At  both  extremes  we  tend  to 
lose  sight  of  the  component  tones.  In  consonance  they  run  too  much 
into  one  another,  in  dissonance  they  obscure  and  confuse  one  another 
too  much. 

Such  a  view  would  not  quite  coincide  with  the  useful  grading  of 
fusion  from  a  maximum  to  a  minimum  on  the  basis  of  unitariness. 
But  that  would  be  no  insuperable  barrier.  We  might  still  conserve  this 
grading  as  a  partial  aspect  of  the  problem  and  at  the  same  time  prefer 
the  other  as  more  adequate  to  the  sensory  stuff.  Loss  of  distinction 
in  unity,  balance  of  distinction,  and  loss  of  distinction  in  confusion  can 
certainly  be  logically  represented  as  a  series  from  a  maximum  to  a 
minimum — as  a  decrease  from  approximation  to  the  balance  of  a  single 
tone, — and  therefore  valid  for  scientific  purposes.  But  for  musical 
purposes  the  other  notion  which  centres  upon  the  point  of  balance  of 
distinction  seems  by  far  the  more  important. 

For  it  simply  lays  upon  our  hands  the  solution  of  the  problem  of 
the  great  and  fundamental  importance  of  the  thirds  and  sixths  in  all 
music  and  of  the  triad  in  modern  music.  The  thirds  and  sixths  are  the 
intervals  of  greatest  balance  of  distinction  of  tones.    Two  or  more 

10—2 


148  COMMON  CHORDS  OR  CONCORDANCE  [ch. 

thirds  or  sixths  after  one  another,  therefore,  also  afford  as  easy  distinction 
as  one.  Hence  we  pass  immediately  to  the  interpretation  of  this 
distinction  as  melodic  distinction.  And  so  a  series  of  thirds  or  sixths 
is  most  favourable  to  melodic  continuity,  as  the  whole  system  of  facts 
gathered  together  in  the  previous  chapters  have  shown  us  to  be  the 
case. 

The  importance  of  the  triad  for  music  therefore  lies  in  the  two 
thirds  it  contains.  And,  of  course,  an  alternative  is  created  by  the  two 
possible  positions  of  the  major  and  minor  thirds  in  each  common  chord. 
Thus  we  get  the  major  chord  for  the  one  and  the  minor  chord  for  the 
other.  Two  other  possibilities  exist,  namely  the  triads  containing  two 
minor  thirds  or  two  major  thirds.  These,  however,  each  contain  another 
important  intervaL  The  tritone  of  the  one  is  a  distinct  dissonance. 
The  (augmented  fifth  or)  minor  sixth  of  the  other  is  not  ordinarily  a 
dissonance,  but  a  neutral  interval,  a  maximal  balance  of  distinction. 
But  in  the  triad  it  always  acts  as  a  dissonance.  Many  reasons  might 
be  suggested  for  this.  We  need  not  attempt  to  find  the  most  probable 
one  at  this  point.  Having  been  carried  thus  far  by  both  fact  and  logic 
we  may  claim  to  recognise  as  fact  that  the  fundamental  triads  both 
contain  a  fifth  between  the  outer  tones  of  their  two  thirds.  Upon 
these  two  triads  all  harmony  is  said  to  revolve.  But  the  other  two 
triads  also  are  in  common  use.  The  distinctive  feature  of  the  common 
chords  is  due  to  the  fifth  they  contain.  This  high  grade  consonance 
gives  the  whole  a  special  unitariness  or  stability;  but  this  aspect  of 
things  we  shall  leave  for  special  treatment  in  a  later  chapter. 

The  same  principle  that  explains  the  essence  of  the  common  triads 
will  account  also  for  the  discords  that  form  so  important  a  part  of 
modern  music.  If  two  neutral  intervals  may  be  combined  to  form  a 
triad,  it  follows  as  a  matter  of  course  that  three  or  more  may  be  combined 
to  form  greater  chords.  These  will  always  be  discords.  For  the  repetition 
at  the  octave  of  any  one  of  the  components  of  a  common  triad  gives,  as 
we  have  seen,  merely  an  extension  of  the  characteristic  whole  or  pattern 
formed  by  the  three  essential  tones.  Thus  we  obtain  a  set  of  chords 
in  which  all  the  possibilities  of  combination  of  thirds  may  be  exhausted. 
Many  other  ranges  of  possible  combinations  may  be  taken  into  view, 
if  all  the  possible  inversions  of  groups  of  three  thirds  are  examined^. 

The  musical  utility  of  any  of  these  chords  will,  of  course,  depend,  not 

^  The  deduction  of  these  paragraphs  is  not  meant  to  imply  that  only  chords  derived 
from  columns  of  thirds  are  to  be  countenanced.   Of  this  we  shall  see  more  as  we  proceed. 


XVII]  COMMON  CHORDS  OR  CONCORDANCE  149 

so  much  upon  the  neutral  nature  of  the  thirds  that  make  up  what  has 
been  held  to  be  their  original  position,  as  upon  the  kind  of  intervals 
that  are  actually  formed  between  each  pair  of  voices  that  appear  in 
the  chord.  It  would  be  a  mistake  to  take  any  interval  or  class  of  intervals 
as  the  primary  ground  of  a  chord  to  the  disadvantage  or  depreciation 
of  any  other.  All  the  intervals  that  occur  in  a  chord  are  of  equal 
importance,  except  the  octave  (for  the  reason  we  have  given).  When 
a  seventh  or  a  second  occurs,  it  has  the  fusional  status  of  a  second  or 
of  a  seventh,  and  by  no  means  that  of  the  third  that  may  ensue  upon 
its  inversion  or  in  relation  to  some  other  tone  of  the  chord  than  the 
bass  of  the  interval  in  question. 

The  view  we  thus  obtain  of  the  part  played  by  thirds  in  the  establish- 
ment of  chords  falls  into  line  with  the  empirical  principle  that  was 
extracted  by  Rameau  from  the  aesthetic  work  of  music  and  that  has 
been  used  and  defended  repeatedly  since.  This  is  "the  theory  of  the 
generation  of  chords  by  adding  thirds  together"  (60,  8i).  It  has  indeed 
never  been  proved  in  any  sense  of  the  term  (cf.  above,  end  of  chapter  x). 
But  it  has  always  made  a  strong  claim  to  recognition  merely  by  the 
force  naturally  inherent  in  it,  apart  from  all  theory,  as  a  generalised 
expression  of  empirical  practice^.  And  on  this  ground  it  must  be  held 
to  be  far  more  worthy  than  all  the  attempts  to  found  a  system  of  chords 
upon  the  resonance  of  the  sonorous  body  or  upon  the  series  of  partial 
tones. 

Much  time  and  energy  has  been  wasted  upon  the  problem  of  the 
fundamental  chord  or  chords  from  which  all  the  others  are  derived, 
upon  the  systematisation  of  chords  for  the  purpose  of  finding  their 
origin,  and  such  like  questions.  Certainly  it  was  extremely  important 
for  musical  study  to  discover  the  connexions  between  chords  that  we 
know  as  inversion.  That  achievement  is  a  piece  of  direct  and  unshakable 
description,  which  musical  theory  has  to  explain  in  some  way  or  other. 
And  it  was  equally  valuable  to  work  out  the  differences  between  systems 
of  intervals  and  chords,  that  are  summarised  in  the  distinction  of  major 
and  minor  modes.  But  it  is  as  absurd  to  put  one  chord  down  as  the 
origin  of  another,  as  it  would  be  to  consider  a  single  interval,  or  a  single 
tone  as  the  one  and  only  progenitor  of  all.   Besides  any  chord  whatever 

»  Well  expressed  by  M.  H.  Glyn  (18,  sn):  "The  third  has  always  been  beloved  by  the 
natural  ear.  We  have  to  deal  here  with  a  fact  of  far  greater  importance  to  music  than  any 
in  the  science  of  acoustics,  and  if  consonance  to  music  means  the  third,  and  only  in  a 
limited  degree  the  fifth  and  the  fourth,  while  to  science  it  means  the  fifth  and  the  fourth 
and  after  that  the  third,  it  is  clear  that  two  points  of  view  are  being  named  by  the  same 
name  which  are  by  nature  different  and  should  bo  recognised  as  such." 


150  COMMON  CHORDS  OR  CONCORDANCE        [ch.  xvii 

can  be  reduced  to  a  column  of  thirds  by  suitable  transposition  of  its 
tones  through  octaves.  That  follows  from  the  fact  that  a  continuous 
column  of  thirds  (major  or  minor)  soon  yields  all  the  tones  of  the 
chromatic  scale.  Thus,  c,  e,  g,  b,  d},f^,  a^,  c^  —  the  diatonic  scale;  the 
notes  of  the  chromatic  scale  can  be  got  by  suitable  substitution  of 
minor  for  major  thirds  and  conversely. 

A  musical  experimentalist  is  free  to  form  any  interval  he  can  upon 
his  instrument.  If  it  is  good  and  useful  he  will  introduce  it  wherever  it 
will  produce  or  enhance  a  desired  effect.  But  he  must  above  all  make 
it  possible  for  the  listener  to  hear  it  properly;  and  that  enforces  the 
limitation  of  the  number  of  chords  and  of  their  positions  in  the  tonal 
range.  The  whole  history  of  music  is  an  attempt  to  find  a  system  of 
tones  which  will  yield  the  greatest  variety  of  chords  and  the  greatest 
number  of  relations  between  them  that  in  turn  will  most  facilitate  the 
apprehension  of  the  tones  played  and  make  possible  the  greatest  scope 
and  freedom  of  aesthetic  effects. 

It  is  easier  and  more  natural  to  bear  in  mind  the  actual  development 
of  music  from  the  earliest  times  and  to  see  how  the  science  of  musical 
sounds  develops  towards  more  and  more  complete  explanation  of  its 
course  than  it  is  to  try  to  derive  music  from  an  unintelligible  genealogy 
of  its  final  products.  The  musical  mind  of  the  world  did  not  begin 
under  the  inspiration  of  a  subconscious  appreciation  of  all  musical 
effects.  It  began  with  a  very  limited  sense  or  feeling  for  these  things. 
But  enjoying  what  it  already  had,  it  strove  to  make  that  little  grow  to 
greater  ends.  And  as  it  laboured,  the  effects  were  formed  experimentally 
and  the  ear  seized  them.  The  growth  of  the  art  gradually  revealed 
more  and  more  subtle  aspects  and  wider  and  wider  connexions  or 
systems  of  effects.  Two  of  the  greatest  of  these  are  polyphony  and 
tonaUty.  These  were  not  given;  they  had  to  be  discovered.  Art  is  as 
much  a  process  of  discovery  as  science  is.  Both  are  experimental  and 
systematic.  But  while  art  is  content  to  be  empirical,  science  is  restless 
till  it  has  grasped  the  whole  system  of  inner  bonds  that  rule  its  objects 
and  has  described  them  fully  and  thoroughly. 


CHAPTER  XVIII 

MELODIC  MOTION  IN  RELATION  TO  DEGREES  OF  CONSONANCE 

The  analysis  of  the  degrees  of  fusion  we  have  just  given  may  be  extended 
in  a  way  that  seems  to  be  of  importance. 

We  have  distinguished  three  chief  grades  :  loss  of  distinction  in 
unitariness,  balance  of  distinction,  and  loss  of  distinction  in  confusion. 
And  we  have  noticed  that  balance  of  distinction  must  make  for  ease 
and  continuity  of  melody,  when  two  or  more  melodies  run  side  by  side. 
The  question  that  now  promises  to  further  our  insight  into  the  structure 
of  music  is  :  what  effect  has  loss  of  distinction  upon  melodic  continuity, 
upon  the  ease  and  distinctiveness  with  which  two  or  more  melodies 
will  run  side  by  side? 

High  grade  consonance  we  have  already  learnt  to  look  upon  as 
balance,  approximation  to  the  unity  of  a  single  tone.  Its  component 
tones  are  wrought  together  more  than  usual;  they  cling  together  and 
do  not  offer  to  pass  with  as  much  ease  into  two  other  tones  as  they 
were  approached  from  those  that  preceded  them.  They  constitute, 
therefore,  a  point  of  relative  rest  and  tend  to  bring  the  voices  to  a  stop. 

The  same  effect  is  produced  under  certain  circumstances  by  the 
unison,  as  we  have  seen  above  (p.  130).  A  unison  is,  of  course,  from  an 
absolute  acoustical  point  of  view  a  single  tone,  certainly  not  an  interval. 
And  a  single  tone  is  not  as  such  in  music  arrestive  in  function.  Unison 
has  a  significant  meaning  only  in  so  far  as  two  melodic  streams  are 
felt  to  meet  and  to  be  identical  in  a  certain  sound.  Being  usually  two, 
they  then  'sound  one.'  When  such  a  unison  is  presented  without 
correction  by  the  various  circumstances  brought  to  bear  upon  high 
grade  consonances  to  make  them  mobile,  it  produces  the  same  undesirable 
effect  as  they  do.  Two  voices  are  caught  up  into  one  and  an  effect  of 
arrest  and  confusion  is  produced  by  the  loss  of  distinction  in  unity.  It 
is  only  in  this  way  that  the  functional  similarity  of  unison  and  high 
grade  consonances  can  be  justified. 

This  function  of  stability  and  arrest*  peculiar  to  the  consonances 

of  the  octave  and  fifth  has  long  been  recognised  and  shows  itself  in 

various  ways.   In  Greek  and  early  Western  music  the  octave  (or  unison) 

was  the  usual  close  of  a  piece.    The  root  position  of  a  triad  is  a  more 

*  Or  'repose':  cf.  D.  F.  Tovey,  74  passim. 


152  MELODIC  MOTION  IN  RELATION  [ch. 

stable  form  than  either  of  its  inversions;  the  fifth  in  it  spans  the  two 
thirds  or  gives  at  least  (in  the  alternative  arrangement)  a  fifth  with  the 
bass.  In  the  first  inversion  there  is  either  no  fifth  at  all  or  in  the  alternative 
arrangement  only  between  the  two  upper  voices.  (Conjunction  with  the 
bass  we  have  already  seen  to  be  generally  more  powerful  than  conjunc- 
tion with  the  soprano.)  In  the  second  inversion  we  find  the  peculiar 
feature  of  a  fourth  from  the  bass.  As  a  consonant  interval  that  will 
produce  some  effect  of  rest  corresponding  to  its  grade,  which  approxi- 
mates towards  the  neutral  range,  though  still  above  it.  Of  these  three 
forms  of  the  common  chord  the  first  is,  as  the  older  theorists  said, 
*  most  apt  to  conclude ' ;  it  produces  the  greatest  arrestive  effect  upon 
the  flow  of  voices.  When  this  effect  is  heightened  by  special  means — by 
the  use  of  the  chief  transitions  of  tonality,  dominant  or  subdominant 
to  tonic,  and  by  suitable  rhythmical  exposure,  etc.,  we  get  the  various 
cadences,  whose  sole  function  is  to  produce  partial  or  complete  arrest. 
Other  contributory,  and  therefore  functionally  similar,  conditions  are 
a  gradual  reduction  of  speed,  a  greater  sonorousness  and  steadiness  in 
tone  production,  the  repetition  of  the  final  chord,  and  so  on. 

The  counterpart  of  the  high  grade  consonances  is  formed  by  the 
high  grade  dissonances.  These  create  a  loss  of  distinction  in  confusion, 
which  must  also  have  a  disturbing  effect  upon  the  ease  and  continuity 
of  melody.  The  treatment  of  dissonances  in  music  is  a  natural  con- 
sequence of  this.  If  dissonances  are  to  be  introduced,  special  means 
must  be  employed  to  overcome  the  loss  of  distinction  as  far  as  possible. 
Devices,  such  as  suspension  and  preparation,  were  early  invented  and 
rigorously  prescribed.  Although  they  are  not  now  considered  to  be 
indispensable,  they  have  by  no  means  been  superseded  as  superfluous. 
They  serve  to  fix  in  advance  in  the  hearer's  mind  the  most  difficult 
part  of  the  group  of  tones  about  to  be  presented,  whereupon  the  others 
are  introduced  by  common  and  easy  melodic  procedure.  In  this  way 
the  listener  is  enabled  to  follow  the  movement  of  all  the  melodies  equally 
well,  and  so  is  shielded  from  the  confusion  that  might  otherwise  arise. 
Besides  this  subtractive  method  of  approaching  dissonances,  there  is 
of  course  the  method  of  contrary  motion  by  which  the  tones  forming 
a  dissonance  with  one  another  may  be  approached  from  opposite  sides. 
The  listener  is  thus  guided  carefully  through  the  moment  of  confusion. 

The  musical  mind  of  to-day  has  grown  so  familiar  with  all  the 
melodic  combinations  our  harmonic  procedure  has  reduced  to  distinct 
types,  that  there  are  many  who  almost  suggest  that  in  the  course  of  time. 


xvm]  TO  DEGREES  OF  CONSONANCE  153 

as  music  progresses,  what  was  previously  discord  comes  to  be  reckoned 
as  concord.  Stumpf  said:  "Effect  upon  feeling  is  specially  liable  to 
change,  even  within  our  system,  in  that  the  unpleasantness  of  discords 
weakens  and  through  the  introduction  of  new  and  ever  bolder  discordant 
structures  the  old  ones  take  on  the  feeUng  effects  of  concords;  so  that, 
as  v.  Hornbostel  remarked,  progressions  to  these  old  discords  can  act 
soothingly  like  a  resolution  into  concords"  (71,  341 1.).  Stumpf,  however, 
does  not  think  that  such  changes  of  feeling  will  ever  break  down  the 
difference  between  concord  and  discord.  We  may  well  agree  with  him; 
for  a  definite  reason  can  be  given  that  seems  to  be  of  substantial  weight. 

However  familiar  we  may  become  with  the  patterns  of  discords, 
that  will  surely  never  in  any  way  alter  the  graded  differences  there  are 
between  consonances,  neutral  fusions,  and  dissonances  in  respect  of 
balance  of  tonal  distinction.  Both  the  current  theory  of  the  derivation 
of  all  intervals  from  the  series  of  partials  and  Stumpf's  theory  of  fusion 
grade  the  intervals  in  a  series  of  indefinitely  decreasing  consonance. 
The  one  end  of  Stumpf's  series,  as  we  have  seen,  is  characterised  by 
apparent  unitariness  of  sound,  the  other  by  closer  and  closer  approxima- 
tion to  mere  clear-cut  apprehension  of  two-ness.  The  theory  of  partials 
suggests  that  the  nearest  and  loudest  and  perhaps  most  frequent 
partials  yield  the  consonances  that  are  distinguished  earliest  in  the 
history  of  music,  and  that  as  music  advances  our  familiarity  with 
partials  extends  farther  along  the  series,  so  that  we  reckon  as  consonances 
always  as  much  as  we  have  thus  made  our  own.  The  history  of  music 
seems  to  provide  a  parallel  to  this  in  the  early  use  of  the  octave,  the 
subsequent  'organising'  in  fifths  and  fourths,  and  the  later  classification 
of  thirds  and  sixths,  or  even  sevenths  as  consonances.  But,  on  our 
interpretation,  all  this  line  of  speculation  is  completely  cut  out.  The 
series  of  fusions  has  its  neutral  point — or  it«  region  of  natural  ease  and 
familiarity,  as  it  were — in  the  middle,  in  the  thirds  and  sixths.  From 
this  point  the  difficulty  of  manipulating  the  intervals  in  polyphony 
or  in  harmonic  music  increases  in  the  two  opposite  directions — towards 
the  consonances  and  towards  the  dissonances.  Familiarity  may  give  us 
greater  facility  in  dealing  with  these  naturally  recalcitrant  intervals; 
it  may  even  induce  us  to  dispense  with  certain  aids  to  apprehension 
that  we  once  found  necessary  or  desirable.  But  it  cannot  alter  the 
natural  differences  between  the  various  grades. 

Once  we  have  found  the  true  system  of  functions  of  intervals,  the 
false  motive  suggested  by  the  apparent  course  of  history  entirely  loses 
its  value.    The  historical  order  of  approach  is  quite  irrelevant  and  can 


154  MELODIC  MOTION  IN  EELATION  [ch. 

be  readily  explained  otherwise.  It  was  the  prevalence  of  monophony 
that  led  to  the  adoption  of  the  octave  first  of  all  intervals ;  monophony 
does  not  essentially  change  in  becoming  homophony.  And  homophonies 
in  fifths  and  fourths  are  the  next  inevitable  attempts  at  continuous 
development.  The  great  consonances  call  early  attention  to  themselves. 
But  it  is  only  in  polyphony  that  the  polyphonic  functions  of  these  and 
all  the  other  intervals  can  be  discovered.  And  it  is  these  functions  of 
intervals — whereby  they  either  yield  a  simultaneity  of  easy  flowing 
melodies  or  disturb  one  another  in  this  respect — that  determine  the 
final  classification  of  intervals. 

As  for  the  plea  that  the  Greeks  had  not  yet  recognised  the  consonance 
of  the  thirds  and  sixths,  the  evidence  seems  to  indicate  merely  that 
they  did  not  reckon  these  intervals  among  the  distinct  consonances. 
Neither  do  we  really.  They  do  not  show  a  notable  degree  of  approxima- 
tion to  unity  ("so  that  the  resulting  sound  is  one  like  and  similar  to  a 
single  one,"  Nicomachus — 71,329;  66,54),  That  still  leaves  room  for 
two  other  classes,  one  in  which  the  two  sounds,  far  from  being  one- 
like, are  rather  specially  two-like, — shall  we  say? — or  diaiphomc^, 
(discordant,  contraposed  ("when  the  sound  of  the  two  is  as  it  were 
rent  asunder  and  without  true  blending,"  Nicomachus — 66,  54);  and 
another  middle  one  in  which  the  two  sounds  are  just  two,  neither 
friends  nor  enemies,  but  just  comrades. 

Knowledge  by  acquaintance  may  change  then ;  and  so  may  knowledge 
by  theory,  and  practice,  and  familiarity  and  all  such  adjuncts  of  feeling 
or  sensory  experience ;  but  sensory  feeling  itself  does  not  seem  to  change. 
The  conformations  of  sense  retain  their  characters  unaltered.  Sense 
is  a  stufE  that  the  growing  mind  of  man  may  learn  to  mould  as  he  can, 
but  ever  in  obedience  to  the  laws  inherent  in  it.  It  is  as  much  an  objective 
world  that  we  must  learn  to  know  and  to  use  as  is  the  world  of 
nature. 


*  Cf.  14,  96:  "La  sensation  auditive  produite  par  les  consonnances  et  les  dissonances 
est  analysee  d'une  manifere  uniforme  par  tons  les  ecrivains:  "dans  la  consonnance  les 
deux  sons  se  m^langent  au  point  de  s' absorber  mutuellement,  de  telle  manifere  que  I'oreille 
ne  re9oive  qu'une  impression  imique,  douce  et  suave."  Elien  le  platonicien  compare  la 
consonnance  k  "du  vin  mele  de  miel,  ou  aucune  des  deux  substances  ne  predomine,  et 
ayant  le  gout  d'un  breuvage  particulier,  qui  n'est  ni  du  miel  ni  du  vin.  Dans  la  dissonance, 
au  contraire,  le  melange  ne  s'op^re  pas;  les  sons  se  repoussent,  pour  ainsi  dire,  Vxm  1' autre, 
et  I'impression  totale  est  dure  et  penible."  Of  course  the  idea  in  "ni  du  miel  ni  du  vin" 
must  not  be  pushed  to  the  extreme  of  positing  a  new  resultant  third  tone  (cf.  16,  issf.; 
66,  52).  "Aussi  les  definitions  antiques  de  la  symphonic  et  de  la  diaphonie  sont-elles  en 
grande  partie  sanctionees  par  roreille  modeme"  (16,  iss). 


xvm]  TO  DEGEEES  OF  CONSONANCE  155 

But  to  pass  on.  Dissonance  must  differ  from  consonance  not  only 
in  the  way  described,  but  in  another  subtle  manner.  If  consonance 
creates  the  effect  of  a  pause  or  rest  by  presenting  us  with  an  approxima- 
tion to  imity,  and  if  we  accept  the  suggestion  to  rest,  or  if  it  does  not 
conflict  with  the  effect  of  the  other  tendencies  of  sound  at  the  moment, 
but  forms  a  consistent  whole  with  them,  then  we  shall  be  somewhat 
careless  of  distinguishing  differences  within  the  unitariness.  We  are  then 
either  wholly  or  relatively  at  rest  and  we  do  not  need  to  be  scrupulous 
in  distinguishing.  We  have  no  need  for  such  finished  distinctions,  for 
we  do  not  crave  to  move  forwards.  If,  on  the  other  hand,  we  want 
the  music  to  move  forwards  through  a  consonance,  in  spite  of  the 
tendency  of  consonance  to  create  repose,  we  must  be  careful  not  to 
strengthen  the  reposeful  effect  by  any  means  of  the  same  tendency. 
Hence  flow  the  rules  against  consecutive  octaves  and  fifths,  and  the 
exposure  of  these  intervals. 

In  dissonances,  on  the  contrary,  a  point  of  imrest  is  created.  There 
is  neither  rest  nor  even  flow,  but  a  disruptive  effect,  and  disagreement 
between  the  component  tones.  They  get  in  each  other's  way  and  produce 
mutual  confusion  and  disturbance.  Even  if  we  have  been  led  skilfully 
into  the  dissonance,  we  nevertheless  are  impelled  forwards.  We  look 
for  another  phase  of  progress  in  which  disturbance  shall  cease.  But 
not  any  consonance  (or  dissonance)  may  follow;  only  one  which  we 
can  easily  and  melodically  reach  from  the  present  dissonance.  Thus 
the  need  for  resolution  of  dissonances  arises.  When  several  voices  run 
concurrently,  and  there  is  consequently  more  to  follow  and  more  danger 
of  losing  the  thread  of  sequence,  the  need  for  resolution  is  all  the  greater, 
and  must  be  the  more  carefully  controlled.  Thus  it  appears  that  dis- 
sonance, far  from  being  a  barrier  and  a  hindrance  to  good  music,  acts 
clearly  as  a  stimulant  upon  melodic  activity,  urging  it  forwards  and 
increasing  expectation  of  progress. 

Much  might  be  said  in  favour  of  the  adoption  of  the  common  Greek 
terms  'symphony'  and  'diaphony'  with  the  addition  of  the  rarer 
term  'paraphony.'  In  symphony  the  tones  of  an  interval  tend  to  become 
indistinguishable  through  too  much  unitariness  or  fusion;  in  diaphony 
they  sound  through  or  against  one  another,  disturbing  and  confusing 
one  another;  in  paraphony  there  is  balance,  so  that  melodies  formed 
of  such  intervals  will  flow  evenly  side  by  side,  the  one  not  inhibiting 
the  apprehension  of  the  other.  Paraphony,  it  should  be  noted,  does 
not  imply  that  the  tones  of  such  an  interval  are  on  the  whole  more 


156  MELODIC  MOTION  IN  KELATION  [ch. 

easily  apprehended  as  mere  duality  than  the  tones  of  a  diaphony. 
In  the  latter  the  duality  is  indirectly  emphasised  by  the  harsh  confusion. 
Paraphony  implies  merely  a  riedium  grade  of  obviousness  of  plurality 
through  lack  of  approximation  to  the  unity  of  a  single  tone  (in  the 
sense  of  Stumpf's  fusion),  but  at  the  same  time  a  maximum  grade  of 
distinguishability  for  those  musical  purposes  which  require  distinction 
of  pitches  and  apprehension  of  melodic  flow  in  more  voices  than  one. 
For  these  purposes  we  require  a  perfectly  clear-cut  untroubled  dis- 
tinguishability of  the  component  tones  of  an  interval.  That  is  plainly 
wanting  at  the  consonantal  end  of  the  fusional  series.  It  is  merely 
presumed  to  be  present  at  the  dissonantal  end,  because  at  that  end  there 
is  the  more  obvious  indication  of  two-ness  of  tone;  the  tones  jar  upon 
one  another  harshly  and  there  may  be  very  obvious  beating  between 
them ;  even  the  unmusical  mind  reads  there  signs  confidently  as  two-ness, 
while  the  musical  ear  finds  in  the  dissonances  more  obvious  lack  of 
fusion  or  presence  of  ruption  than  in  the  thirds  and  sixths.  But  it  would 
be  right  to  claim  that  in  the  small  dissonances  it  is  more  difficult  to 
pick  out  the  pitches  of  the  component  tones  than  it  is  in  the  thirds  and 
sixths,  and  that  in  the  larger  dissonances,  although  the  pitches  stand 
well  apart,  yet  the  two  tones  do  'oppose'  one  another  or  jangle  with 
one  another^,  besides  being  very  much  two-like  apart  from  that.  Thus 
the  validity  of  the  fusional  series,  as  we  find  it  in  Stumpf  and  others, 
would  be  called  in  question.  In  music  an  abstraction  of  melodic  function 
from  apparent  one-ness  and  jarring  two-ness  has  been  carried  through, 
which  points  to  the  neutral  thirds  and  sixths  as  the  region  of  pure 
two-ness  of  tone  in  interval.  The  clear  distinction  of  melodies  must 
therefore  be  but  a  seriation  of  this  two-ness  function,  not  a  new  kind 
of  (perpendicular)  distinction  of  tones  supervening  upon  their  melodic 
combination.  It  is  a  task  for  experimental  work  to  find  and  to  describe 
exactly  the  analytic  attitude  that  will  confirm  this  conclusion  from  the 
functions  of  intervals  in  polyphonic  music. 

Our  new  results  may  be  summarised  very  briefly  with  the  aid  of 
these  terms.  The  movement  in  music  is  melodic.  For  the  proper  flow 
of  simultaneous  melodies  intervals  must  either  be  themselves  actually 
paraphonic  or  they  must  be  used  paraphonically. 

Note. — Of  the  Greek  terms  relating  to  harmony,  symphony  and 
diaphony  are  by  far  the  most  familiar  and  have  gone  over  simply 

^  In  particular  the  lower  boundary  of  the  higher  tone  falls  near  to,  and  so  defiles, 
the  symmetrical  outline  of  the  lower  tone  in  its  most  important  point — its  pitch. 


XVIII]  TO  DEGREES  OF  CONSONANCE  157 

into  the  terms  consonance  and  dissonance.  I  have  quoted  characteristic 
examples  of  their  definitions  above  (p.  154).  The  term  paraphony  was 
used  by  several  later  writers,  Thrasyllus,  Bacchius  and  Gaudentius 
(66,  48f.,  67  ff.;  16,  139).  While  the  relevant  passage  in  Bacchius  is  almost 
certainly  confused,  Thrasyllus  attached  the  term  to  the  fifth  and  fourth 
in  distinction  from  the  octave,  a  subdivision  which  does  not  seem  to 
be  of  any  particular  interest,  as  far  as  we  are  concerned  nowadays. 
But  the  relative  passage  in  Gaudentius  is  of  the  greatest  importance. 
It  has  apparently  been  a  source  of  mystification  for  most  interpreters. 
Stumpf  speaks  of  it  as  "this  otherwise  (i.e.  than  by  his  explanation) 
quite  incomprehensible  passage"  (66,  72)i. 

Since  my  conclusions  call  so  clearly  for  the  use  of  a  term  like  '  para- 
phony,' it  becomes  a  matter  of  much  interest  to  consider  whether 
Gaudentius  may  have  been  led  to  his  somewhat  similar  application  of 
the  same  term  from  a  similar  train  of  thought.  Apart  from  the  mere 
existence  of  the  term,  its  verbal  meaning,  and  the  conceptual  setting 
of  the  notion  between  symphony  and  diaphony,  I  can  find  Mith  the 
help  of  the  chief  authorities  (Gevaert  and  Stumpf)  little  or  nothing  to 
show  what  the  idea  of  Gaudentius  really  was.  Stumpf  suggests  merely 
that  Gaudentius  meant  the  major  third  and  the  tritone  to  be  taken 
simply  as  "consonances  of  lower  grade,  as  transition  to  the  dissonances" 
(66,  70). 

Stumpf's  translation  of  the  passage  in  Gaudentius  is  as  follows  : 

Symphonic  are  those  in  which,  when  they  are  simultaneously  struck  or  blown 
on  the  flute,  the  melos  of  the  lower  in  relation  to  the  higher  or  conversely  is  always 
the  same,  or  (in  which)  as  it  were  a  fusion  in  the  performance  of  two  tones  occurs 
and  a  kind  of  unity  results.  Diaphonic  are  those  in  which  when  they  are  simul- 
taneously struck  or  blown,  nothing  of  the  melos  of  the  lower  in  relation  to  the 
higher  or  conversely  appears  to  be  the  same  or  which  show  no  sort  of  fusion  in  relation 
to  one  another.  Paraphonic  are  those  that,  standing  in  the  middle  between  the 
symphonic  and  the  diaphonic,  yet  appear  symphonic  when  played;  which  seems 
to  be  the  case  in  the  tritone  (f-b)  and  the  ditone  (g-b),  (66,  69). 

Elsewhere  (66,  66)  Stumpf  says  that  melos  is  "perhaps  best  translated 
as  'the  melodic  element  of  tone'  or  the  tonal  element  of  melody.'* 

No  doubt  the  matter  might  be  discussed  at  great  length.  But  if 
we  leave  the  first  clauses  of  Gaudentius's  definitions  as  obscure,  and 
(or)  interpret  them  by  the  following  clauses,  then  we  must  look  upon 
Gaudentius's  use  of  the  term  'paraphony'  as  well  founded.  Symphony 
is  the  greatest  fusion  (when  the  melos  of  the  two  tones  tends  to  sameness), 

*  Cf.  14,  09:  "Mais  11  importe  de  remarquer  que  cette  doctrine  est  Isolde  dans  la  littdra> 
tore  musicale  des  anciens." 


158  MELODIC  MOTION  IN  RELATION  [ch. 

diaphony  is  the  least  fusion  (when  the  melos  of  the  two  tends  to  extreme 
difference),  paraphony  is  the  middle  between  these,  which  appears 
as  symphony  in  performance.  It  is  probably  this  last  clause  that  has 
compelled  writers  to  think  Gaudentius  meant  this  middle  class  to  be 
a  lower  grade  of  consonance,  not  a  really  middle  neutral  class,  and  so 
to  miss  perhaps  the  main  point  of  Gaudentius's  distinction.  That  main 
point  is  the  placing  of  the  neutral  relation  (para)  in  the  middle  of  the 
whole  series  and  the  attaching  of  it  in  particular  to  the  major  thicd. 
It  is  much  more  significant  than  any  distinction  between  the  octave 
and  the  fifth  with  the  fourth.  Music  naturally  gives  the  octave  a  special 
place,  because  it  is  its  unit  of  division,  whereby  repetitions  in  octaves 
become  practically  mere  repetitions  or  identities.  But  the  grading  of 
octave,  fifth,  fourth  had  been  estabhshed  long  before  Gaudentius  (by 
Aristoxenus,  66,  38).  The  former  did  not  add  the  major  third  to  them 
as  a  lower  grade  of  consonance,  but  as  a  member  of  another  class  lying 
between  the  consonances  and  the  dissonances. 

It  may  seem  perverse  to  stress  the  point;  but  it  is  really  an  important 
one.  Music  makes  a  distinction  of  opposition  between  consonance  and 
dissonance,  but  it  has  failed  to  recognise  that  distinction  theoretically 
in  so  far  as  it  ranks  the  thirds  and  sixths  as  imperfect  consonances,  not 
as  neither  consonances  nor  dissonances  but  neutral  'sonances'  or 
paraphonies.  Gaudentius  makes  this  division,  adding  the  note  that 
paraphonies  sound  symphonic  in  (instrumental)  performance  (iv  ry 
Kpovcrei). 

In  a  sense,  our  present  musical  classification  may  be  said  to  be 
just  the  obverse  of  the  early  Greek  one,  primitive  and  limited  as  that 
has  been  usually  thought  to  be  by  modern  writers.  The  older  Greeks 
made  one  dividing  line  below  the  fourth,  calling  all  the  rest  diaphonies. 
We  draw  our  great  line  below  the  thirds  and  sixths,  calUng  all  except 
the  dissonances  consonances.  Let  us  combine  the  two  and  we  get — 
in  principle — the  divisions  of  Gaudentius.  And  if  we  hke,  we  can  add 
with  him  that  ev  rfj  Kpovaev — in  a  mere  interval  of  two  tones  struck 
together  as  distinct  from  two  melodies  (as  we  may  say) — the  paraphonies 
seem  symphonic.  In  other  words  when  it  is  not  a  matter  of  paraphony 
generally,  but  just  of  the  general  character  of  a  chord  as  a  whole,  we 
reckon  the  specific  paraphonies  (thirds  and  sixths)  to  the  symphonies. 
Or,  we  might  say  :  we  take  them  not  as  apparently  one-like  like  the 
high  grade  consonances,  but  as  agreeing  with  one  another  and  therefore 
as  consonant,  because  they  do  not  obviously  jar  upon  one  another 
as  do  the  distinct  dissonances. 


XVIII]  TO  DEGREES  OF  CONSONANCE  159 

Of  course  it  would  be  improper  to  read  into  Gaudentius  all  that  we 
can  now  put  into  the  skeleton  of  the  distinctions  to  fill  them  out.  But 
he  should  have  the  benefit  of  any  doubt  there  may  be.  His  conceptual 
scheme  is  full  enough,  but  there  is  not  enough  detail. 

All  modern  theorists  have  treated  Gaudentius's  distinction  as  if  it 
merely  amounted  to  an  extension  of  the  grades  within  the  class  of 
consonances  by  a  further  step  downwards.  That  surely  does  violence 
to  his  words  and  to  his  term  (paraphony).  If  this  modern  view  is  wrong, 
then  with  it  must  go  the  attempt  to  see  an  evolution  of  the  notion  of 
consonance  downwards  from  the  octave,  to  include  first  the  fifth  and 
fourth,  then  the  thirds  and  sixths,  now  the  natural  seventh  and  tritone, 
and  to-morrow  all  the  dissonances  themselves  (cf.  22,  115).  Impossible! 
That  were  no  evolution,  but  a  debasement.  Evolution — unless  it  be 
the  degeneration  of  the  parasite  that  casts  off  its  sense-organs — means 
progress,  an  increase  in  the  complexities  or  in  the  differences  distin- 
guished, not  the  swamping  of  all  differences  in  one  class^.  All  differences 
remain  as  they  were  given,  but  we  learn  to  know  them  and  their  functions 
better,  and  to  use  them  practically  in  our  art  without  feeling  shocked  or 
lost  amongst  the  more  refractory  ones^. 

^  So  the  distinctions  made  by  such  a  writer  as  Johannes  de  Garlandia  show  rather  a 
keen  sense,  than  a  'parade'  (81,  im)  of  scientific  accuracy.  His  series  is^  unison,  octave; 
6,  4;  3,  III;  VI,  7;  U,  6;  2,  T,  VII.   Cf.  6,  vol.  i.  104 f. 

*  It  may  be  of  some  objective  interest  to  note  that  I  made  a  reference  to  Gaudentius 
on  pages  15,  108  above,  but  at  that  time  I  saw  (like  all  other  students  of  the  subject, 
I  suppose)  nothing  specially  significant  in  the  term  paraphony.  In  fact  I  had  in  the 
meantime  till  writing  the  present  section  forgotten  the  existence  of  this  term.  The  train 
of  thought  expounded  above  made  me  feel  the  need  for  a  term  to  cover  the  range  between 
consonance  and  dissonance,  but  different  from  either.  Latin  is  here  insufficient  unless 
we  say  sonance;  so  I  put  together  the  term  paraphony.  Months  afterwards  on  turning 
over  the  pages  of  Gevaert  I  was  astonished  and  delighted  to  see  the  word  paraphony 
there.  The  point  of  interest  is  that  whether  unconscious  cerebration  was  agog  in  this 
coincidence — which  I  greatly  doubt — or  not,  the  objective  differences  analysed  and  dis- 
cussed demanded  the  term;  it  was  not  suggested  by,  or  transferred  from,  Gaudentius. 
So  perhaps  my  analysis  will  do  him  a  good  turn  for  the  dignity  he  gives  to  the  term  I 
chanced  upon. 


CHAPTER  XIX 

MELODY  (OR  PARAPHONY)  AS  THE  PRIMARY  BASIS  OF  MUSIC 

We  have  now  reached  a  point  of  view  from  which  we  can  survey  a  large 
part  of  the  realm  of  music. 

The  view  we  obtain  has  been  clearly  indicated  in  the  thesis  which 
forms  the  title  of  this  chapter.  Melody  is  the  primary  basis  of  all  music. 
By  melody  we  mean  a  special  phenomenon  of  motion  or  passage  between 
two  tones  that  appear  before  a  mind  in  successive  moments  separated 
from  one  another  by  a  certain  interval  of  time  which  may  vary  in 
size  within  certain  limits  under  various  conditions.  The  successive  tones 
must  not  be  so  different  (in  loudness  and  blend)  from  one  another  as 
to  appear  to  come  from  different  sources  and  so  to  suggest  an  objective 
independence  of  one  another.  That  circumstance  is  unfavourable  to 
melodic  connexion.  But  it  is  not  our  present  concern  to  study  the 
nature  of  the  motion  involved  in  melody,  or  in  short  of  melody  in 
general.  We  have  here  taken  melody  for  granted,  as  a  famiUar  pheno- 
menal fact.  The  reader  is  supposed  to  know  already  what  melodic 
connexion  is,  as  he  surely  does,  being  able  to  tell  at  once  whether  the 
'  passage '  from  one  note  to  another  is  there  or  whether  there  is  a  break, 
a  suspension  of  motion  or  passage,  as  there  is  for  example  after  a  close 
or  a  half  close,  etc.  (For  an  account  of  the  primary  theoretical  study 
of  melody  in  this  sense,  see  77,  chapter  vi.) 

Melody,  in  short,  is  the  motion  of  music. 

But  the  word  is  often  used  to  mean  more  than  that,  namely  the 
series  of  pitches  through  which  a  melodic  motion  passes.  For  general 
theoretical  purposes  it  is  best  to  use  for  this  a  word  that  hnks  this 
feature  of  sound  to  the  analogous  feature  in  the  other  senses.  When  a 
visual  motion  passes  through  a  number  of  points,  we  say  it  marks  out 
a  certain  form  or  figure.  Thus  a  burning  torch  swung  quickly  round 
leaves  a  trail  that  forms  more  or  less  of  the  circumference  of  a  circle. 
A  flying  meteor  marks  out  a  straight  line,  and  so  on.  We  may  say 
similarly  that  a  melody  falls  into,  or  has,  a  certain  form  or  figure.  The 
notion  is  quite  familiar  to  musical  literature.  Perhaps  the  word  '  theme ' 
and  its  derivatives  '  thematic '  and  '  thematised '  are  less  open  to  confusion 
with  heterogeneous  subjects  than  are  any  of  the  other  words  that  bear 
a  similar  meaning,  such  as  tune,  motive,  subject,  etc. 


CH.  xix]  THE  PRIMAKY  BASIS  OF  MUSIC  161 

If  we  use  the  term  'melody'  for  the  general  notion  of  motion  from 
note  to  note,  then  we  may  divide  melodies  into  two  classes — those 
that  are  thematic  or  that  show  a  definite  form  capable  of  coherent 
analysis,  and  those  that  are  not  thematic.  In  the  latter  there  may  be 
plenty  of  motion  from  tone  to  tone,  i.e.  plenty  of  melody  in  general, 
but  as  little  form  as  there  is  in  the  motion  of  a  fly — to  take  a  homely 
instance.  The  fly  is  always  on  the  move,  and  so  is  a  dancer.  But  there 
is  figure  or  form  in  the  dancer's  actions,  while  in  the  fly's  there  is  practically 
none. 

In  this  sense  melody  is  the  primary  basis  of  all  music. 

With  80  general  a  meaning  in  the  term  melody,  this  statement 
partakes  very  much  of  the  nature  of  a  truism.  Music  undoubtedly 
began  as  melody.  Apart  from  purely  rhythmic  art  in  which  sound 
plays  only  the  part  of  a  practically  unvaried  medium,  all  early  music 
is  simply  melodic.  And  it  is  thematic  in  a  more  or  less  simple  way  as 
well.  But  it  is  generally  supposed  to  have  ceased  to  be  wholly  or 
thoroughly  melodic  at  a  certain  point  of  its  development  in  Europe,  and 
to  have  become  harmonic;  and  in  so  changing  to  have  struck  into  a 
new  line  of  development  that  was  present  only  in  minute  traces,  if  at 
all,  in  primitive  or  in  ancient  music.  This  new  line  has  led  to  wonderful 
forms  of  art,  unshadowed  and  undreamt  in  the  first  origins  of  music. 
Harmony  seems  to  be  a  new  creation  within  music;  a  new  dimension, 
one  might  say ;  the  perpendicular  complement  to  the  horizontal  functions 
of  melody,  as  has  been  said. 

No  doubt  harmony  has  come  to  be  of  great  importance.  It  is  not 
easy,  however,  to  say  precisely  what  its  scope  is.  But  there  seems 
little  doubt  that  the  melodic  functions  of  music  have  been  considerably 
underestimated.  In  fact,  from  the  point  of  view  of  theory,  harmony 
has  usually  been  put  down  as  the  one  and  only  basis  of  true  music. 
A  music  in  which  harmony  is  evidently  neither  implicit  as  in  polyphony, 
nor  explicit  as  in  harmonic  music,  hardly  deserves  the  name  of  art. 
It  is  merely  primitive  play,  as  it  were. 

An  almost  contrary  thesis  may  be  vigorously  maintained.  It  may 
be  claimed  that  melody  is  the  primary  and  continuous  basis  of  music; 
remaining  so  even  throughout  harmonic  developments,  which  are 
essentially  a  by-product  of  melodic  complexity,  always  carefully  sub- 
ordinated to  the  prior  and  essential  requirements  of  melodic  movement. 

In  the  earliest  music  there  is  only  one  line  of  melody,  only  one  voice. 
Harmony  shows  itself  at  most  only  in  the  reduplication  of  this  melody 

w  r.  u.  11 


162  MELODY  (OR  PARAPHONY)  AS  [ch. 

at  the  octave,  fifth,  or  fourth,  or  in  an  irregular  accompaniment  of 
intervals  to  the  tones  of  the  melody  which  does  not  make  a  second 
melody  and  does  not  seem  to  interfere  with  the  apprehension  of  the 
one  distinct  melody.  From  this  somewhat  chaotic  state  the  art  of 
polyphony  in  two  and  then  in  three  or  more  voices  gradually  emerges 
in  that  the  accompanying  tones  take  on  the  form  of  a  distinct  voice 
chiefly  by  the  device  of  contrary  motion  with  the  chief  voice^.  Then 
the  art  of  polyphony  becomes  clearly  conscious  and  slowly  attains  a 
sense  of  the  true  principles  of  style  suitable  to  simultaneous  melodies 
(cf.  48,  chapter  ii).  The  rules  regarding  consecutive  and  exposed 
intervals  are  then  gradually  discovered.  These  rules  are  necessary  in 
order  that  the  melodic  distinction  of  the  voices  may  be  clear  and  easy. 
The  arrestive  effect  of  the  'symphonies'  is  of  the  greatest  importance 
where  arrest  of  melodic  progression  is  desired ;  but  it  must  be  carefully 
avoided  when  melodies  have  to  flow  easily  together.  Diaphonies  must 
not  lead  to  the  confusion  of  melodies;  we  must  be  led  safely  through 
them  and  on  to  groupings  of  tones  that  the  ear  can  readily  apprehend. 
But  there  is  in  this  kind  of  art  no  difference,  except  that  of  difficulty, 
between  the  construction  of  two  simultaneous  melodies  and  of  three 
or  more.  Harmony  does  not  necessarily  become  explicit  with  the 
grouping  of  three  voices;  nor  does  music  in  two  voices  (now)  necessarily 
fail  to  be  apprehended  harmonically;  there  are  many  who  claim  that 
even  a  single  melody  is  necessarily  apprehended  harmonically,  absurd 
though  that  claim — in  the  face  of  primitive  music — must  be  held  to 
be.  Of  course  polyphony  can  hardly  begin  to  be  apart  from  a  cogently 
melodic  or  even  thematic  treatment  of  the  combined  melodies.  That 
follows  naturally  from  their  very  being  as  distinct  melodies.  There 
would  be  as  little  interest  or  beauty  in  the  erratic  or  merely  melodic 
motion  of  several  voices  as  there  would  be  for  us  in  a  song  made  up  of 
a  more  or  less  random  succession  of  tones  2. 

^  It  was  Helmholtz  that  suggested  that  "the  first  of  such  examples  could  scarcely 
have  been  intended  for  more  than  musical  tricks  to  amuse  social  meetings.  It  was  a  new 
and  amusing  discovery  that  two  totally  independent  melodies  might  be  sung  together 
and  yet  soimd  well"  (20,  244).  But  the  actual  course  of  development  must  have  been 
more  natural  and  continuously  meaningful  than  that. 

2  C  V.  Stanford  (62,  6)  extends  this  idea  in  a  general  rule  with  considerable  emphasis: 
"Mere  combinations  of  notes,  in  themselves  sounding  well,  but  without  logical  connexion 
with  their  successors,  are  useless  as  music.  The  simultaneous  presentation  of  two  melodies 
which  fit  each  other  is  at  once  a  musical  invention;  and  when  a  third  or  fourth  melody 
is  added  to  the  combination,  the  result  is  what  is  called  harmony.  To  speak  of  studying 
harmony  and  counterpoint  is,  therefore,  to  put  the  cart  before  the  horse.  It  is  coimterpoint 
which  develops  harmony  and  there  is  no  such  boundary  wall  between  the  two  studies 
as  most  students  imagine."   Our  conclusions  re-affirm  this  last  statement. 


XIX]  THE  PRIMARY  BASIS  OF  MUSIC  163 

Out  of  polyphony  harmony  was  gradually  abstracted  by  a  slow 
process  of  association  of  simultaneous  tones  and  familiarisation  with 
their  perpendicular  aspects.  The  lines  of  melodic  motion  had  to  meet 
often  and  often  ere  the  patterns  of  their  contacts  could  become 
thoroughly  known.  And  still  longer  time  was  required  before  men 
found  that  these  patterns  could  be  ranged  in  succession  in  most  fascinating 
ways.  This  origin  of  harmony  is  in  a  sense  quite  familiar.  It  has  been 
well  expressed  by  Sir  Hubert  Parry  in  his  articles  in  Grove's  Dictionary 
of  Music  and  Musicians.  At  the  end  of  the  article  on  Harmony  he 
speaks  of  it  summarily  as  "the  principle  that  harmony  is  the  result 
of  combined  melodies." 

The  ecclesiastical  cadences  were  nominally  defined  by  the  progressions  of  the 
individual  voices,  and  the  fact  of  their  collectively  giving  the  ordinary  Dominant 
Cadences  in  a  large  proportion  of  instances  was  not  the  result  of  principle,  but  in 
point  of  fact  an  accident.  The  Dominant  Harmonic  Cadence  is  the  passage  of  the 
mass  of  the  harmony  of  the  Dominant  into  the  mass  of  the  Tonic  "  (46,  sio). 

But  harmonic  music  has  not  ceased  to  be  essentially  melodic.  The 
thematisation  characteristic  of  polyphony  has  disappeared  from  the 
majority  of  the  voices  perhaps;  but  each  chord  must  still  be  connected 
with  the  preceding  in  ways  which  make  melodic  connexion  easy  and 
which  do  not  allow  its  clear  flow  to  be  arrested  by  excessive  symphony 
or  dissipated  by  extreme  diaphony.  The  paraphonic  components  of 
chords  are  perhaps  their  essential  musical  constituents.  And  the 
discovery  of  new  chords  may  be  said  to  be  the  development  of  new 
paraphonic  combinations.  Most  of  the  modern  systems  of  chord  forma- 
tion have  emphasised  the  importance  of  the  superposition  of  thirds  as 
a  principle  of  origin.  We  can  now  understand  it  not  merely  as  an 
empirical  principle,  but  as  a  principle  of  function  (in  so  far,  of  course, 
as  it  rests  not  upon  a  mere  arithmetic  of  thirds  by  transposition  through 
octaves,  but  upon  a  naturally  felt,  and  so  real,  connexion  through  the 
similarities  of  inversions  to  one  another  as  patterns).  The  primary  need 
is  not  so  much  a  wealth  of  consonantal  and  dissonantal  effects,  as  a 
clear  and  intelligible  flow  of  melody  through  such  harmonic  effects  as 
are  compatible  therewith. 

"By  the  use  of  chromatic  passing  and  preUminary  notes,"  Sir  Hubert  Parry 
says,  "  by  retardations,  and  by  simple  chromatic  alterations  of  the  notes  of  chords 
according  to  their  melodic  significance,  combinations  are  arrived  at  such  as  puzzled 
and  do  continue  to  puzzle  theorists  who  regard  harmony  as  so  many  unchangeable 
lumps  of  chords  which  cannot  be  admitted  in  music  unless  a  fundamental  bass 
can  be  found  for  them"  (46,  sis;  cf.  47,  and  48,  2S4). 

11—2 


164  MELODY  (OR  PARAPHONY)  AS  [ch. 

This  interpretation  of  the  development  of  chords  is  to  be  preferred 
to  Stumpf's.  It  is  certainly  not  necessary  to  introduce  the  notion  of 
concordance  in  order  merely  to  explain  the  conjunction  of  more  voices 
than  two.  The  notions  of  consonance,  or  better,  of  paraphony — as  may 
perhaps  be  said  generically — are  quite  adequate  to  the  use  of  intervals, 
not  only  between  two  voices,  but  also  between  more  than  two.  No 
change  of  basis  or  of  principle  is  thereby  required.  The  essential  basis 
of  music  ever  was  and  remains  melodic  movement.  One  melody  is 
self-contained;  simultaneous  melodies  must  be  mutually  compatible. 
Single  melodies  are  almost  inevitably  thematic;  simultaneous  melodies 
may  be  so  too;  but  the  thematisation  of  one  or  more  may  be  dropped 
in  favour  of  the  interest  created  by  their  harmonic  fusion.  Or,  finally, 
as  D.  F.  Tovey  expresses  it,  modern  melody  may  be  merely  (is)  "the 
surface  of  a  series  of  harmonies "  (75)^.  When  harmony  emerges, 
however,  it  is  not  a  new  creation;  it  is  still  precisely  the  same  thing 
as  is  the  fusion  or  balance  of  two  tones,  except,  of  course,  that  it  is 
of  greater  scope  and  detail  in  several  voices  than  in  two.  It  has  now 
only  been  made  the  centre  and  object  of  the  artist's  creative  genius. 
The  art  has  become  perpendicular  in  build  instead  of  horizontal. 

But  the  rejection  of  harmony  or  concordance  as  an  essentially  new 
element  in  polyphony  does  not  imply  that  the  latter  in  its  harmonic 
form  has  brought  forth  nothing.  There  has  certainly  been  great  develop- 
ment and  growth,  so  much  indeed  as  to  create  striking  differences  in 
the  styles  of  art.  The  interest  created  by  these  was  so  great  that  much 
of  what  had  been  toilfully  gained  in  polyphonic  art  was  temporarily 
abandoned  and  fell  into  common  neglect.  That  must  perhaps  always 
happen  to  any  art  when  new  constructive  vistas  appear. 

There  seems  to  be  no  doubt  that  of  the  acquisitions  of  the  new 
art  the  most  important  and  fundamental  was  the  principle  of  tonality. 
In  polyphony  the  thematisation  of  all  the  melodies  held  them  artistically 
together.  This  common  structure  could  only  be  abandoned  when  a  new 
principle  of  connexion  had  become  apparent,  something  that  would 
link  the  various  voices  together  throughout  the  changes  of  their  harmonic 
patterns.  These  patterns  had  to  be  wrought  into  an  intelligible  system, 
and,  as  we  know,  even  the  outline  of  the  scales  had  to  be  altered  to 
make  this  possible. 

^  Or,  "modem  melody  is  the  musical  surface  of  rhj^hm,  harmony,  form,  and  instru- 
mentation. In  short  melody  is  the  surface  of  music."  This  may  be  the  modem  culmination 
of  melody,  but  it  is  certainly  a  wrong  definition  of  melody  in  general,  including  primitive 
melody. 


XIX]  THE  PRIMARY  BASIS  OF  MUSIC  165 

It  is  tonality  that  gave  and  continues  to  give  the  chief  impulse  to 
the  systematisation  of  chords.  These  are,  and  perhaps  can  only  be, 
systematised  in  relation  to  tonality,  and  its  three  poles — tonic,  dominant 
and  subdominant.  It  is  a  perversion  of  actual  history  to  suppose  that 
chords  were  first  derived  from  the  mere  tone  (fundamental  and  partials), 
bringing  implicit  in  them  the  determinants  of  tonality  and  of  our 
scales.  Scales  were  developed  long  before  either  tonality  or  chords 
had  been  conceived  or  even  felt.  On  the  contrary  the  course  portrayed 
in  history  seems  systematically  much  more  acceptable.  First  came  mere 
motion  in  tones  (without  any  implicit  scale)  or  mere  melody;  out  of 
melody  was  begotten  by  the  force  of  familiarity  and  the  needs  of  social 
co-operation  scale;  scale  made  further  complications  and  co-operations 
possible,  leading  to  polyphony;  the  habits  and  traditions  of  polyphony 
engendered  harmony  and  tonality;  and  that  as  it  grew  reflected  upon 
its  progenitors  and  moulded  them  to  its  better  development;  it  adjusted 
the  scale  to  its  systematic  requirements  and  reduced  the  functions  of 
thematised  melody  in  favour  of  shorter  bonds  of  melody  running 
through  and  combining  large  tonal  masses.  Thus  line  was  reduced  in 
favour  of  mass;  but  the  mass  itself  is  now  also  treated  linearly,  as  it 
were;  or  it  is  made  to  move  round  a  central  axis  of  orientation.  It  is 
only  after  this  time  that  we  find  the  first  beginning  of  a  systematic 
exposition  of  chords  in  Rameau.  And  we  must  remember  that, 
curiously  enough,  one  of  the  first  steps  in  this  work  was  the  identification 
of  those  groups  of  tones  that  we  know  as  inversions  of  one  another. 

We  have  already  discussed  the  problem  of  inversions  and  have 
introduced  the  notion  of '  volumic  pattern '  to  account  for  their  connexions 
and  differences.  Postibly  the  chordal  aspect  of  modern  music  is  founded 
upon  this  attitude  towards  volume,  whereby  the  group  of  tones  is  con- 
sidered rather  as  a  whole  than  as  so  many  stages  in  so  many  melodies 
of  which  the  themes  are  clearly  held  in  mind.  Of  course  the  mind  has 
still  to  be  led  melodically  through  these  masses.  But  the  interest  lies 
not  in  the  forms  created  by  the  melodic  movements  as  such,  but  in  the 
masses,  their  patterns  or  '  surfaces '  or  '  colours '  (as  some  say)  into  which 
the  mind  has  been  easily  and  safely  led.  The  modern  interest  would  lie, 
then,  not  so  much  in  the  forms  of  all  the  motions  through  which  the 
listener  is  carried,  as  in  the  phases  or  masses  of  sound  in  which  he  from 
moment  to  moment  finds  himself. 

The  canvas  is  now  filled  with  broader  effects.  These  are  still  aft 
built  upon  lines  of  movement  (i.e.  melodies),  but  the  lines  are  now 
subdued  and  hardly  appear  from  a  distance,  so  to  speak,  except  it 


166  MELODY  (OR  PARAPHONY)  AS  [ch. 

may  be  in  the  one,  or  perhaps  two,  that  run  through  the  mass  in  its 
changes  and  give  its  movements  thematic  form  and  beauty. 

Tonality  may  be  said  to  be  perhaps  the  broadest  of  these  effects. 
In  its  earliest  forms  it  is  merely  a  centre  or  point  for  the  harmonic 
stream.  In  its  later  and  freer  developments  it  might  almost  be  considered 
as  a  new  form  of  movement — a  new  melody — of  all  the  (unthematised) 
voices  at  once  and  it  builds  itself  up  like  the  primitive  scale  upon  the 
simplest  consonances,  of  which  the  fifth  is,  of  course,  the  first  distin- 
guishable one, — the  octave  being  by  its  function  already  mere  identity 
or  repetition.  So  we  get  the  poles  of  tonality, — the  dominant  and 
subdominant.  And  these  again  give  others,  equally  closely  allied  to 
them,  until  a  whole  system  appears  through  which  the  harmonic  stream 
may  be  made  to  wander  for  its  greater  diversity  and  beauty. 

One  of  the  perennial  problems  of  tonality  has  been  the  nature  and 
origin  of  the  difference  between  the  major  and  the  minor  scales.  Their 
actual  constituents,  as  we  have  noted,  have  been  determined  by  the 
systematic  requirements  of  the  groupings  of  the  moving  voices  that 
are  most  essential  to  them.  In  their  most  rudimentary  form  these 
reduce  entirely  to  the  two  thirds  (and  sixths) — the  two  chief  inherently 
paraphonic  intervals.  The  tonal  difference  between  the  major  and 
minor  keys  may  then  be  read  from  their  symbols  of  'origin'  in  the 
two  common  chords  ceg  and  ce\)g. 

The  different  relations  of  these  two  to  the  partials  of  their  supposed 
root  c  have  been  grossly  overrated  in  importance.  Certainly,  if  c  has  a 
full  series  of  lower  partials,  e  and  g  will  coincide  with  them.  But,  as 
has  so  often  been  pointed  out,  the  partials  of  e  and  g, — which  cannot 
be  ignored, — confuse  the  issue.  For  e  gives  g^  and  6,  while  g  gives 
h  and  d.   As  Macfarren  said  : 

It  is  of  course  necessary  for  practical  musical  purposes,  not  only  to  make  a 
selection  of  notes  from  the  endless  harmonic  series,  but  to  confine  the  use  of  harmonics 
to  those  belonging  to  certain  exceptional  generators,  or  roots,  in  every  key;  otherwise 
every  note  of  every  chord  might  be  supposed  to  furnish  its  harmonic  series,  and 
each  of  these  its  harmonics  in  turn,  all  sounds  would  confuse  all  other  sounds, 
tonality  would  be  at  an  end,  and  Babel  would  reign  supreme  (35,  94). 

In  spite  of  the  fancies  of  recent  composers  in  search  of  new  scales 
(23,  72f.,  and  'passim;  cf.  5,  24),  these  harmonic  origins  have  for 
many  a  long  year  obviously  been  as  dead  as  a  door-nail.  A  much 
simpler  origin  of  such  latest  varieties  is  their  actual  source — the  chro- 
matic scale  of  just  or  of  equal  temperament,  consciously  treated  by  the 


XIX]  THE  PRIMARY  BASIS  OF  MUSIC  167 

process  of  'aesthetic  selection'  that  Helraholtz  emphasised.  No  doubt, 
however,  the  coincidence  of  partials  will  make  for  slight  differences 
of  smoothness;  these  we  have  already  admitted  as  modifications  of 
smoothness  otherwise  given. 

Paraphonically  the  two  common  chords  show  practically  no  difference 
at  all.  Both  contain  a  major  and  a  minor  third  and  a  fifth.  The  major 
third  is  'exposed'  in  the  major  chord  by  its  resting  on  the  bass  voice; 
the  minor  third  is  similarly  'exposed'  in  the  other.  Thus  the  two 
chords  are  practically  equivalent. 

The  only  difference  that  remains  is  the  difference  of  pitch  of  the 
middle  tone.  It  lies  lower  in  the  minor  chord  than  in  the  major.  And 
this  difference  must  run  equally  throughout  the  whole  of  the  two  scales. 
Any  composition  distorted  from  its  major  tonality  into  the  tonic  minor 
differs  from  the  former  solely  in  the  lowering  of  all  the  mediants  and 
submediants  by  a  semitone.  The  minor  form  differs  from  the  major 
by  an  inner  lowering  of  pitch.  It  is  more  '  voluminous,'  heavier,  darker, 
sadder,  etc.  Change  to  the  major  key  means  a  streak  of  lesser  volumes, 
a  brighter,  lighter,  clearer  atmosphere,  as  it  were^. 

When  the  problem  is  thus  cleared  of  its  false  and  artificial  difficulties, 
the  solution  is  easy  and  it  appears  to  be  quite  natural  and  inevitable. 
There  is  no  special  agony  of  harmonic  birth  in  the  minor;  we  do  not 
look  upon  the  major  from  below  and  upon  the  minor  from  above; 
they  are  not  the  mirror  images  of  one  another.  The  minor  tonality 
partakes  merely  of  that  difference  that  appears  in  the  raising  of  the 
pitch  niveau  of  a  composition.  Smaller  volumes  suggest  brighter, 
lighter  effects;  or  they  merely  are  smaller,  more  precise,  tones,  apart 
from  all  suggestion.  And  so  the  minor  key  appears  woven  with  one 
larger  thread  throughout ;  our  minds  respond  to  this  auditory  difference 
with  any  analogously  varied  experiences  we  may  have  ready  in  our 
memories  and  moods^. 

It  is  a  familiar  fact  that  in  recent  years  many  experiments  have 

^  Cf.  the  detailed  empirical  investigation  of  Becker's,  where  it  is  shown  that  "Dur 
bedeutet  schlechthin  Lust,  moll  schlechthin  Unlust"  (2,  aw);  and  (p.  258):  "As  expressions 
of  pleasure  in  the  minor  are  not  free  of  unpleasant  moments,  while  on  the  contrary  major 
as  expression  of  displeasure  approximates  by  a  greater  or  less  indefiniteness  of  coloration 
to  the  minor  character,  and,  besides,  the  displeasure  value  of  many  contents  is  not  very 
distinct,  therefore  we  can  uphold  the  meaning  of  the  modes  as  opposite  feeling-tones  of 
corresponding  musical  devices." 

*  Contrariwise  and  similarly,  for  the  ancient  Greeks  "the  passage  of  the  minor  third 
into  the  major  expressed  a  lowering,  a  depression"  (16,  S24).  For  in  their  music  the  melody 
lay  below  the  accompaniment,  so  that  the  major  third  (reckoned  downwards),  gave  the 
touch  of  weight  or  sadness.   On  thirds  in  Greek  music  sec  also  Iti,  9M. 


168  MELODY  (OR  PARAPHONY)  [ch.  xix 

been  made  by  composers  towards  the  establishment  of  new  schemes 
of  melody  and  harmony.  It  is  unnecessary  to  specify  or  to  describe 
these  here.  Time  must  first  show  what  is  their  capacity  for  lasting 
artistic  usage.  But  it  is  evident  that  the  basis  of  music  expounded  in 
the  preceding  chapters  is  quite  ready  and  able  to  accommodate  all  that 
is  proved  to  be  acceptable  in  them.  If  our  theoretical  analyses  and 
constructions  correspond  fully  to  the  chief  long  tried  and  approved 
forms  of  music,  they  will  apply  to  the  newer  departures  in  their  degree 
of  success.  For  these  actually  hold  already  for  the  musical  mind  a 
very  definite  relation  to  the  earlier  established  music.  We  hear  and 
know  what  we  gain  in  the  new  and  what  we  lose  in  it  that  made  the 
old  precious.  There  is  in  all  a  gradual  transition  and  development. 
The  mind  may  grow  familiar  with  the  natural  and  artistically  achieved 
paraphonies  of  the  main  drift  of  musical  art  and  tire  of  them.  It  may 
then  strive  to  curb  the  lesser  paraphonies  to  its  will.  But  the  gain  of 
novelty  in  chords  of  fourths  must  involve  a  great  strain  upon  the 
apprehension.  Many  groupings  of  lesser  paraphonies  must  be  in  actual 
effect  much  more  diaphonic  than  paraphonic,  discordant  more  than 
melodious.  But  it  is  not  our  present  intention  to  defend  or  to  justify 
any  one  of  such  experiments,  but  merely  to  show  that  the  basis  of 
analysis  and  theory  already  offered  not  only  leaves  room  for  them, 
but  can  even  anticipate  the  losses  and  deficiencies  they  are  likely  to 
entail.  That,  as  already  said,  can  be  done  without  any  theory,  merely 
from  the  analytic  foundations  of  harmony,  as  it  is  generally  known. 
The  theory  given  above  merely  shows  how  these  effects  are  based  in 
the  actual  auditory  stuff  itself. 

Of  course,  only  the  fringe  of  this  vast  subject  has  been  touched. 
One  of  the  great  difficulties  that  face  the  theory  of  music  in  general  at 
the  present  time  is  that  the  art  is  so  highly  developed,  while — in  spite 
of  the  vast  amount  of  analytic  work  that  has  been  done  in  connexion 
with  the  form  and  structure  of  music — the  theory  of  the  basis  of  music 
in  the  auditory  stuff  of.  tones  has  hitherto  been  really  non-existent. 
Reaching  the  fringe  of  the  subject,  therefore,  implies  much  more  than 
at  first  appears.  We  have  not  yet  explained  much.  For  the  detail  of 
that  work  the  centuries  remain.  But  we  have  at  least  now  a  fairly 
clear  view  of  the  promised  land,  in  search  of  which  men  have  wandered 
so  widely  and  aimlessly. 


CHAPTER  XX 

THE  FACTORS  THAT  MODIFY  PARAPHONY 

We  may  now  use  the  term  paraphony  to  indicate  not  only  the  specifically 
paraphonic  intervals  of  thirds  and  sixths,  but  also  all  intervals — 
symphonic  as  well  as  diaphonic,  specially  consonant  or  dissonant, — 
in  so  far  as  they  are  made,  or  become,  paraphonic,  or  in  so  far  as  their 
actual  or  potential  paraphonic  function  is  capable  of  modification  by 
various  factors.  These  factors  we  have  already  to  some  extent 
encountered  in  the  previous  chapters,  and  shall  now  proceed  to  gather 
together  and  to  resolve  as  far  as  possible  into  their  essential  functions. 
We  have  not  only  to  indicate  the  effect  of  each  factor,  but  as  far  as 
possible  also  to  explain  how  that  effect  is  attained. 

We  notice,  then,  that  paraphony  diminishes  from  its  optimum  in 
the  thirds  and  sixths  in  two  directions :  in  the  one  towards  the  sym- 
phonies, which  create  a  loss  of  distinction  in  unity  and  thereby  a  point 
of  relative  balance  and  rest  of  tonal  mass;  in  the  other  towards  the 
diaphonies,  which  make  a  loss  of  distinction  in  confusion  and  thereby 
a  point  of  relative  restlessness  and  propulsion  of  the  tonal  mass. 

We  are  also  familiar  with  the  nature  and  ground  of  the  varied 
exposure  or  ease  of  distinction  of  the  various  pairs  of  voices  :  in  the 
following  decreasing  order, — B-S,  B-A  and  B-T,  S-A  and  S-T,  and 
A-T  (cf.  above,  p.  102  f.). 

The  effect  of  an  increase  in  the  number  of  voices  is  largely  due  to 
the  mere  spreading  of  the  attention.  It  is  more  or  less  a  general  rule 
of  sensory  apprehension  that  the  larger  the  number  of  distinct  items 
to  be  observed  simultaneously,  the  less  distinct  is  each  one,  and  the 
less  easily  is  it  separated  from  the  others  and  observed,  especially  in 
the  field  of  hearing,  where  fusion  and  overlapping  of  volumes  play  so 
important  a  part.  The  various  pairs  of  voices  will,  of  course,  still  have 
the  same  relative  grades  of  exposure,  the  outer  voices  being  the  most 
exposed  and  the  innermost  ones  the  least  so.  But  as  the  maximum 
grade  will  now  be  lower  than  it  was,  the  minimum  will  be  doubly  so; 
for  the  larger  number  of  voices  makes  a  larger  number  of  grades. 
Probably  the  difference  between  these  grades  is  also  smaller.  Thus 
none  of  the  melodies  will  be  quite  so  cogent  in  general,  unless  one  or 


170  THE  FACTORS  THAT  MODIFY  PARAPHONY  [ch. 

other  is  made  specially  prominent  by  increased  intensity  or  by  specially 
distinctive  or  obtrusive  tone-blend  (as  for  example  in  orchestral  music). 
The  effect  of  symphonies  or  diaphonies  upon  the  floAv  of  melody  will 
also  be  less  marked.  Any  slight  disturbance  is  hardly  noticeable  in 
the  mass. 

Besides  any  symphony  or  diaphony  is  now  likely  to  be  separated 
by  several  voices,  so  that  the  unity  that  might  otherwise  be  arrestive 
will  now  be  variegated,  and  the  clashing  that  might  be  confusing  will 
be  well  spread  out  and  manageable.  Even  if  the  thread  of  two  voices 
is  temporarily  lost,  no  great  risk  will  be  incurred;  for  there  is  much 
else  to  engage  the  attention,  and  the  harmonic  patterns  of  the  whole 
chordal  masses  will  still  prevail. 

Contrariwise,  when  the  number  of  parts  is  decreased,  what  is  allowable 
in  four  parts  will  be  governed  by  apparently  stricter  rules.  The  decrease 
in  the  number  of  voices  makes  each  more  prominent,  so  that  the  rules 
for  it  partake  more  of  the  stringency  of  the  rules  for  outer  voices.  At 
the  same  time  the  innermost  voices  fall  away,  and,  as  the  rules  for 
these  are  the  most  lax  of  all,  the  general  average  of  strictness  is  apparently 
raised. 

The  next  question  is  one  of  the  most  important  and  very  puzzling 
in  its  own  way,  unless  it  is  faced  with  the  resolution  of  rigorous  logic  : 
what  is  the  precise  nature  and  basis  of  the  effects  of  similar  and  contrary 
motion? 

Oblique  motion  must,  of  course,  not  be  forgotten.  But  the  answer 
for  it  is  obvious  :  one  of  the  voices  stands  still  and  is  thus  already 
melodically  prepared  for  the  attention,  so  that  the  other  moving  voice 
has  the  greater  freedom  and  scope  so  far  as  the  analytic  attention  is 
concerned.  In  other  words  the  melodic  spontaneity  or  force  of  the 
moving  voice  is  a  matter  of  its  own  coherence  and  expressiveness; 
and  thus  the  melodically  blurring  effect  of  any  symphonies  and 
diaphonies  that  may  occur  between  the  two  voices  is  obviated.  The 
early  use  of  oblique  motion  in  connexion  with  dissonances  is  familiar. 
Similarly  we  have  noticed  how  an  otherwise  unstable  melody  has  more 
cogency,  and  may  even  progress  by  leap,  if  the  other  tones  in  the  chord 
remain  unchanged,  or  if  the  two  successive  chords  differ  only  in  position. 

With  regard  to  similar  motion,  it  is  exceedingly  tempting  to  interpret 
it  as  a  positive  influence.  Its  treatment  in  all  textbooks  of  harmony 
suggests  this  strongly.  The  recent  notion  of  'exposure'  of  octaves 
and  fifths  (58,  268 f.)  has  greatly  increased  the  force  of  this  suggestion. 


XX]  THE  FACTORS  THAT  MODIFY  PARAPHONY  171 

But  that  indication  seems  to  be  very  misleading.  For  similar  motion 
has  to  be  kept  in  relation  with  aU  the  intervals.  And  it  is  important 
to  notice  that  it  has  no  apparent  effect  upon  paraphonies  at  all.  This 
would  compel  us  to  infer  that  its  effect  upon  the  symphonies  and 
diaphonies  is  also  nought;  it  merely  allows  them  to  stand  untouched, 
relieving  neither  voice  of  the  necessity  for  being  cogently  melodic  and 
paraphonic.  This  the  diaphonies  and  still  more  the  symphonies  fail  to 
be ;  for  while  the  former  urge  us  on  in  spite  of  the  confusion  they  tend  to 
produce,  the  latter  arrest  the  melodic  flow — a  most  pernicious  effect  when 
it  is  sufficiently  '  exposed '  by  the  prominence  of  the  voices  that  bear  it. 

The  difference  between  one  symphony  or  diaphony  and  consecutives 
is  then  simply  the  difference  between  one  bad  effect  and  two  of  them  in 
succession^.  One  has  only  to  listen  to  consecutive  fifths  for  a  while  to 
notice  that  the  bad  effect  that  appears  in  them  comes  out  not  merely 
and  solely  when  a  second  one  is  played,  but  that  it  attaches  even  to  a 
single  one.  The  fifth, — and  to  a  less  extent  the  fourth, — is  a  bad  interval 
for  polyphony  in  general,  i.e.  for  paraphony.  No  doubt  it  sounds  well 
as  a  merely  momentary  or  isolated  mass  of  sound,  or  so  long  as  we 
think  of  it  as  detached  from  all  melodic  flow  or  sequence.  But  it  is 
commonly  recognised  to  be  (otherwise)  bare  and  poor.  The  octave  will, 
of  course,  be  still  worse  as  an  interval  than  is  the  fifth,  in  so  far  as  it 
is  heard  as  an  interval.  That  happens  really  only  in  polyphony.  When 
it  is  played  alone,  we  tend  rather  to  apprehend  it  in  its  (then  best) 
musical  function, — merely  as  a  reinforced  single  tone. 

Similar  motion,  therefore,  i&  a  negative  condition.  The  positive 
force  must  rather  be  contrary  motion.  That  will  have  the  effect  of 
favouring  in  all  cases  a  distinction  of  the  voices, — a  distinction  both 
in  symphonies  and  in  diaphonies  and  a  greater  distinguishability  in 
paraphonies.  Contrary  motion,  in  other  words,  favours  the  paraphonic 
effect  all  round. 

The  only  variant  on  these  conclusions  that  might  be  offered  is  the 
assumption  of  a  certain  degree  of  blurring  and  confusing  effect  in  the 
case  of  similar  motion.  The  degree,  however,  would  have  to  be  small 
enough  not  to  affect  the  thirds  and  sixths  disadvantageously.  This 
modification  would  still  leave  the  theory  on  the  whole  identical  with 
the  inferences  stated  above.  In  both  forms  similar  motion  would  rank 
as  a  relatively  negative  factor. 

•  This  is  suggested  by  Shinn  (58,  387)  in  one  instance  when  he  says:  "The  fact  that 
we  admit  consecutive  fifths  when  so  formed  [in  connexion  with  discords],  obviously 
covers  the  admission  of  exposed  fifths  similarly  formed." 


172  THE  FACTORS  THAT  MODIFY  PARAPHONY  [ch. 

It  may  seem  to  be  an  enigma  in  this  connexion  how  it  comes  that 
the  relations  of  motion  towards  a  chord  that  is  not  yet  sounded  makes 
the  analysis  of  that  chord  easier  or  harder,  seeing  that  there  is  no 
continuity  or  slur  of  sound  between  this  chord  and  the  previous  one. 
The  objection  is  well  founded.  The  second  chord  falls  upon  us  unawares 
(more  or  less);  many  other  chords  might  usually  have  been  played 
instead.  The  process,  however,  is  grounded  in  the  nature  of  melody 
(in  the  most  general  sense)  as  a  motion  or  phenomenon  of  motion  from 
one  tone  to  another.  A  second  tone  is  a  sort  of  reservoir  or  line  of 
drainage,  into  which  the  residual  (neural)  activity  of  the  first  tone 
runs  ofE  and  discharges,  so  that  the  two  become  linked  together  by  a 
line  (as  it  were)  of  activity  which  can  only  emerge  when  the  second 
tone  has  made  a  place  for  the  first  to  discharge  into.  Paraphonic 
difierences  rest  upon  the  fact  that  these  discharges  can  be  fully  controlled 
only  under  certain  circumstances.  And  the  artistic  use  of  melody 
requires  their  full  control. 

In  contrary  motion,  then,  the  tones  of  the  second  interval  both  lie 
ordinally  outside  or  inside  the  pitch-range  of  those  of  the  first.  The 
motions  between  the  respective  pairs  of  tones  are,  therefore,  easily 
distinguishable,  and  so  we  get  a  paraphony — the  flowing  of  two  melodies 
side  by  side.  In  the  case  of  successive  thirds  and  sixths  this  paraphony 
is  quite  natural.  The  tones  of  these  intervals  are  evidently  just  the 
right  distance  apart  for  easy  melodic  flow  towards  or  away  from  them- 
selves, even  though  another  third  or  sixth  follows.  Similar  motion  does 
not  spoil  this  effect  at  all.  So  in  symphonies  and  diaphonies  the  bad 
flow  of  melody  is  due  to  the  nature  of  the  intervals,  whereby  their 
tones  enter  into  confusion  with  one  another ;  it  is  not  due  to  the  similarity 
of  motion,  at  least  for  the  greater  part. 

We  can  now  see  readily  the  basis  of  the  familiar  rules  that  (melodic) 
parts  should  not  overlap  or  cross  one  another.  If  they  do,  there  is  a 
great  probabihty  of  a  confusion  of  melodic  connexions  and  so  of  faulty 
paraphony.  The  same  applies  to  such  rules  as  :  "If  the  same  note  is 
found  in  two  consecutive  chords,  it  should  in  general  be  kept  in  the 
same  voice,  ...  as  it  will  conduce  greatly  to  the  smoothness  of  the 
part- writing " ;  and:  "Each  part  should  generally  go  to  its  nearest  note 
in  the  following  chord"  (52,  40)i.  But  if  a  difference  of  tone-blend  is 
given,  as  in  choral  or  chamber  music,  the  voices  may  move  more  freely ; 

^  Cf.  a  more  "modem"  expression  (51,  i4):  "There  must  be  a  connecting  link 
between  successive  chords  of  a  note  common  to  each  or  of  one  or  more  parts  moving  within 
the  interval  of  a  tone  or  a  semitone,  the  other  parts  being  free  in  movement." 


XX]  THE  FACTORS  THAT  MODIFY  PARAPHONY  173 

for  the  blend  of  each  will  serve  to  bind  its  component  tones  together 
(cf.  p.  Ill,  above).  It  is  also  apparent  why  a  second  should  not  proceed 
to  a  unison;  for,  without  the  distinguishing  help  of  contrary  motion, 
this  is  just  confusion  worse  confounded.  In  two  successive  unisons 
also  we  tend  to  lose  sight  of  the  intended  duality  of  parts.  But  one 
unison  is  harmless,  since  the  parts  must  proceed  to  it  from  a  good 
paraphony  and  by  contrary  motion,  while,  in  leaving  it,  contrary 
motion  will  again  be  very  frequent  or  an  easy  paraphony  the  objective. 
Thus  the  two  voices  drop  easily  into  the  one  reservoir  and  as  easily 
discharge  towards  the  following  two.  The  one  tone  is  then  really  a 
unison  psychically.  Progression  to  a  unison  by  similar  motion,  however, 
involves  the  crossing  of  parts  and  is  only  tolerable  under  favourable 
circumstances,  as  by  step  in  one  part,  or  from  dominant  to  tonic,  and 
the  like  (cf.  52,  3i),  Relations  of  visual  motion  very  like  these  motions 
of  tone  have  been  experimentally  established  (78). 

The  interests  of  harmony  are  specially  concerned  in  the  next  funda- 
mental problem  :  what  is  the  efEect  of  the  paraphony  of  simultaneous 
intervals  upon  one  another?  Or,  how  do  they  combine  to  a  resultant? 
An  analogous  question  emerged  naturally  in  the  development  of  the 
study  of  fusion  (as  the  approximation  of  two  tones  to  the  impression 
made  by  a  single  tone),  viz.:  what  degree  of  fusion  appertains  to  an 
assemblage  of  three  tones,  and  how  do  the  individual  fusions  of  its 
three  intervals  contribute  to  the  result?  But,  whatever  may  be  the 
value  of  that  enquiry  in  itself,  we  have  already  seen  that  it  is  not  the 
proper  line  of  approach  towards  music.  The  interval  is,  of  course,  in 
a  sense  the  element  of  structure  in  music.  But  our  systematic  inductions 
have  shown  us  that  the  decisive  consideration  in  the  function  of  each 
interval  is  how  the  two  melodies  of  which  it  forms  a  phase  of  conjunction 
flow  together  through  it.  The  analysis  of  music  naturally  reduces  the 
problem  in  the  first  instance  to  a  study  of  pairs  of  melodies.  For  the 
least  grade  of  harm  is  the  mutual  disturbance  of  two  melodies.  But 
there  is  no  reason  why  three  or  more  melodies  should  not  disturb 
each  other.  And  music  may  properly  consider  these  interferences  if 
they  occur  in  typical  forms.  Our  problem  is  merely  an  extension  of 
our  study  from  two  concurrent  melodies  to  three  or  more. 

But  in  thus  declining  the  implications  of  the  theory  of  fusion  and 
in  denying  its  capacity  for  progress  towards  a  proper  theory  of  music, 
we  do  not  bind  ourselves  to  reject  the  serial  arrangement  of  the  intervals 
in  respect  of  something  which  makes  the  one  preferable  to  the  other 


174  THE  FACTORS  THAT  MODIFY  PARAPHONY  [ch. 

(call  it  'greater  consonance,'  if  you  like)  upon  which  the  theory  of 
fusion  is  based.  In  that  arrangement  the  major  third  stands  before  the 
minor  third  and  the  major  sixth  before  the  minor  sixth.  We  shall, 
however,  in  due  course  have  to  look  back  upon  the  theory  of  fusion  or 
of  paraphony  or  of  consonance,  or  whatever  we  may  call  it,  and  attempt 
to  find  a  sufficient  basis  for  the  functions  of  which  we  are  now  gathering 
a  fuller  and  completer  knowledge. 

In  approaching  this  problem  of  summation,  therefore,  we  must 
make  clear  to  ourselves  first  of  all  what  it  is  that  may  be  added  together. 
In  symphony  there  is  a  loss  of  distinction  in  unity  and  a  consequent 
arrest  of  melodic  movement.  The  effect  of  this  loss  may,  as  we  have 
seen,  be  annulled  by  the  use  of  contrary  motion  and  other  devices. 
But  the  unity  of  the  symphony  survives  to  characterise  the  moment  of 
conjunction  of  the  melodies  and  to  give  it  an  aspect  of  unity,  steadiness, 
and  stability,  which  must  not  be  heightened  if  the  music  is  to  move 
smoothly.  In  the  paraphony  there  is  freedom  of  distinction  and  of 
melodic  movement.  In  diaphony  there  is  a  loss  of  distinction  in  confusion 
which  again  may  be  relieved,  as  far  as  approach  to  it  is  concerned,  by 
special  devices  of  an  analytical  tendency.  But  these  of  course  do  not 
annul  the  basis  of  confusion  in  the  stuff  of  the  tones  themselves,  whatever 
it  may  be.  They  only  make  the  conjunction  melodically  serviceable 
and  clear  in  spite  of  its  confusion.  Or  perhaps  it  keeps  in  a  state  of 
tension  and  incompatibility  what  would  otherwise  be  open  confusion. 

Thus  we  see  that  we  can  hardly  expect  to  find  it  possible  to  add 
symphonies  and  diaphonies  to  a  resultant,  as  we  add  positive  and 
negative  quantities  together.  But  there  seems  to  be  no  ground  of 
incompatibility  between  either  of  these  and  the  neutral  paraph  onies. 
And  we  may  safely  infer  that  the  conjunction  of  several  symphonies 
or  of  several  diaphonies  will  produce  an  effect  of  greater  unitariness 
and  arrest  or  of  greater  tension  and  harshness. 

Thus  in  ceg  there  are  two  paraphonies  ce  and  eg  and  one  symphony  eg. 
The  general  character  of  the  chord  is,  of  course,  symphonic,  i.e.  it  is 
paraphony  bound  together  by  symphony.  If  the  c^  above  is  also  given, 
there  are  added  a  paraphony  ec^  and  two  symphonies  gc^  and  cc^.  The 
octave  displaces  the  fifth  in  the  outer  voices  and  the  fourth  appears 
in  a  relation  of  medium  obscurity.  Thus  on  the  whole  the  symphonic 
effect  is  increased.  In  the  chord  hdp-  there  are  two  (minor  third)  para- 
phonies, and  one  diaphony  6/^,  so  that  the  general  character  of  the  whole 
is  diaphonic. 

But  the  different  exposure  given  by  the  different  pairs  of  voices 


XX]  THE  FACTORS  THAT  MODIFY  PARAPHONY  175 

will  make  differences  between  chords  that  consist  of  the  same  intervals. 
We  should  expect  the  symphonic  (or  diaphonic)  effect  to  be  greatest 
when  the  greater  symphony  (or  diaphony)  lies  in  the  outer  voices, 
less  when  it  lies  between  the  soprano  and  an  inner  voice  than  between 
the  latter  and  the  bass^,  and  least  when  it  lies  in  the  inner  voices.  Thus 
ceg  is  a  greater  symphony  then  ce^g,  because  the  major  third  hes  in 
the  former  on  the  bass,  while  in  the  latter  it  lies  on  the  highest  voice. 
Of  the  inversions  of  the  chord  hd}f^,  b<Pf^  is  the  worst  because  the  tritone 
lies  in  the  outer  voices,  fbd^  is  next  because  it  now  lies  on  the  bass, 
and  dfb  the  best  because  here  it  is  obscured  in  the  upper  voices.  The 
arrangement  dbf^  is  even  better;  for,  while  the  tritone  remains  in  the 
same  place,  the  minor  third  now  displaces  the  major  sixth  in  the  outer 
voices,  and  so  increases  the  tendency  towards  symphony  somewhat^. 

Similarly,  of  the  inversions,  ceg  is  the  best,  because  it  contains  the 
two  best  fusing  paraphonies  and  has  the  fifth  in  the  outer  voices; 
gc^e^,  the  second  inversion,  is  the  next  best,  and  egc^,  the  first  inversion, 
the  least  good,  because  the  latter  has  not  only  lesser  fusion  for  each 
paraphony  (3  for  III,  6  for  VI),  but  it  also  has  its  consonance — the 
fourth — in  the  upper  voices.  In  the  minor  chord  the  relations  are 
similar  for  the  fourth,  but  the  other  two  intervals  are  major  in  the 
first  inversion  and  minor  in  the  second.  How  these  sets  of  changes 
balance  out,  is  a  fine  point  for  experiment  to  settle.  Kemp's  experiments 
show  a  balance  of  preferences  with  respect  to  the  comparative  fusions* 
of  the  two  inversions  of  the  minor  chord,  but  a  preponderance  of 
preference  for  the  second  inversion  of  the  major  (29,  207). 

Regarding  the  alternative  arrangements  for  each  inversion  we  may 
infer,  in  harmony  with  musical  practice,  that,  when  other  things  are 
equal,  the  closer  position  is  better  than  the  extended  one.  Thus  ceg  is 
better  than  cge^,  because  in  the  latter  the  fifth  is  not  in  the  outer  voices, 
but  on  the  bass.  But  in  ge^c^  as  against  gce^  we  get  a  minor  sixth  for 
a  major  third,  and  an  eleventh  (now  in  the  outer  voices)  for  a  fourth. 
There  is  the  same  loosening  of  relations  in  ec^g^  as  against  egc^.    But 

*  Cf.  the  rule  stated  by  Kiilpe  and  confirmed  by  Pear's  experiments  that  where  the 
same  intervals  go  to  form  diflFerent  chords,  the  fusional  degree  of  the  chord  is  greatest 
when  the  better  fusing  interval  lies  lower  (29,  209).  Cf.  21,  Pt  2,  §  91:  "The  more 
perfect  concords  ought  to  be  below,  and  the  less  perfect  above,  in  a  chord." 

*  Cf.  35,  49:  "In  this  inverted  form  it  is  classed  among  the  concords  of  the  ancient 
style." 

*  This  is  Kemp's  term  for  the  object  of  preference  in  his  experiments.  We  should 
not  admit  the  implications  of  the  term  in  this  connexion,  of  course,  as  has  already  been 
suflSciently  indicated. 


176  THE  FACTORS  THAT  MODIFY  PAR  APHONY  [ch. 

the  differences  in  these  cases  cannot  be  great,  since  the  changes  do 
not  produce  any  bad  effects,  while  the  connexions  of  pattern  established 
between  original  inversions  and  their  alternative  arrangements  help  to 
bind  them  together  again.  Nevertheless  the  differences  are  certainly 
there,  and  they  call  for  experimental  as  well  as  for  theoretical 
study. 

Where  diaphonies  appear  in  chords  we  find  in  practice  that  "the 
lowest  fusion  contained  in  the  chord  is  most  decisive  "  (29, 209, 244).  That 
chord  is  most  diaphonic  which  contains  the  greatest  diaphony.  The 
more  exposed  it  is,  of  course,  the  worse  will  be  the  effect,  as  we  have 
already  noted  and  exemplified. 

The  greater  decisiveness  of  the  diaphony  appears  again  in  such 
chords  as  the  dominant  seventh.  But  while  the  fifth  does  not  outweigh 
the  two  discordant  intervals  so  as  to  render  the  whole  chord  purely 
paraphonic  and  not  at  all  diaphonic,  yet,  as  we  have  seen,  the  discords 
do  annul  the  symphonic  effect  of  the  fifth.  The  fifth  is  still  a  fifth, 
both  as  interval  and  as  'fusion';  but  the  arresting  confusing  influence 
it  exerts  upon  the  two  streams  of  melody  that  run  through  it,  is  now 
annulled  by  the  presence  of  the  diaphonic  intervals  in  the  chord, 
especially  if  the  fifth  is  not  isolated  in  the  bass  or  soprano.  So  both 
one  fifth  or  a  second  fifth  may  pass  unguarded  in  the  body  of  such  a 
discord. 

The  'root'  position  gb(Pf^  is  better  than /^6£?^,  because  here  the  fifth 
rests  on  the  upper  voice.  In  the  other  two  inversions  the  fifth  becomes 
a  fourth.  Thus  we  see  that  the  'root'  positions  of  chords  are  not  at 
all  due  to  their  derivation  from  the  bass  note  by  any  indirect  process. 
The  ground  of  preference  lies  in  the  chord  itself,  not  in  the  supposed 
partials  of  one  of  its  notes.  If  the  term  'root  position'  is  to  be  retained, 
let  it  be  understood  in  the  sense  that  in  it  the  chord  is,  as  we  hear  it, 
stablest  and  most  nearly  symphonic.  Moreover,  as  this  stability  is 
referred  naturally  to  the  central  pitch  of  the  whole  chord,  i.e.  to  the 
bass,  the  root  position  of  a  chord  will  form  a  specially  good  approach 
to  any  difficulty  in  the  bass  melody,  e.g.  to  a  fourth  on  the  bass. 

It  is  a  notable  fact  that  both  observation  and  general  theory  thus 
place  the  second  inversion  of  the  major  chord  next  to  the  root  position, 
preferring  it  to  the  first  inversion.  There  is  no  escape  from  this  theoretical 
conclusion  so  long  as  the  fourth  is  ranked  as  the  third  grade  of  con- 
sonance. Even  Helmholtz,  whose  theoretical  foundations  were  the 
successive  steps  of  the  harmonic  series,  was  inevitably  led  to  the  same 


XX]  THE  FACTORS  THAT  MODIFY  PARAPHONY  177 

result^  (20, 214).  The  verdict  of  experimental  observation  can  be 
challenged  only  on  the  ground  of  misdirection  of  description.  But  it 
cannot  be  denied  that  a  certain  point  of  view  leads  to  the  ranking  of 
the  second  inversion  after  the  root  position  and  before  the  first  inversion. 

But  while  the  musical  observations  of  all  time  have  readily  admitted 
the  ranking  of  the  fourth  as  a  consonance  immediately  after  the  fifth, 
the  verdict  of  all  polyphonic  music  has  been  equally  in  favour  of  the 
inversion  in  which  the  fourth  is  in  the  upper  voices  and  against  the 
second  inversion.  In  fact  the  latter  conclusion  has  threatened  at  times 
to  swamp  the  other,  though  it  has  never  really  succeeded  in  doing  so 
completely.  The  fourth  obviously  cannot  be  a  dissonance  generally, 
since  it  is  clearly  consonant  apart  from  the  bass  voice.  Some  special 
circumstance  must  be  responsible  for  the  bad  effect  in  the  bass. 

We  have  already  (p.  135  ff.)  discussed  this  question  and  have  noted 
that  its  probable  basis  is  the  proximity  of  the  (major)  third.  The  fourth 
somehow  suggests  this  other  interval  so  strongly  that  it  seems  itself 
to  be  only  a  point  of  transition  to  the  third.  In  thus  seeming  to  call 
for  a  resolution,  the  fourth  indeed  resembles  the  dissonances.  But  the 
resemblance  is  only  accidental,  in  the  logical  sense.  We  know  now  that 
the  essential  feature  of  dissonance  is  not  merely  its  low  grade  of  fusion, — 
for  the  sixths  have  also  a  low  grade;  a  dissonance  shows  also  a  loss  of 
distinction  in  confusion  of  the  two  tones  that  compose  it  or  of  the  two 
melodies  that  pass  through  it.  For  the  fourth,  however,  we  cannot 
claim  any  lower  grade  of  fusion  than  that  appertaining  to  the  fourth 
in  general;  and,  even  if  there  is  a  certain  confusion  in  the  bass  fourth, 
it  is  not  an  internal  confusion  that  produces  merely  a  loss  of  distinction, 
a  mere  blurring  of  what  is  given  in  the  interval;  it  is  a  confusion  that 
is  due  to  the  attraction  of  the  neighbouring  third;  the  confusion  has 
an  external  reference  beyond  the  interval  actually  given. 

It  is  plain  to  ordinary  musical  observation  that  a  fourth,  exposed 
not  only  by  standing  upon  the  bass,  but  rhythmically  as  well,  calls 
for  the  third  on  the  same  bass,  and  that,  if  this  call  is  to  be  suppressed, 

^  Helmholtz  referred  the  effect  of  the  bass  fourth  to  the  disturbing  effect  of  tonality. 
But  the  tonic  does  not  necessarily  come  into  question  at  all.  On  the  contrary  the  six-four 
chord  earliest  and  oftenest  admitted  is  that  of  the  common  chord  on  the  tonic,  next  is 
that  on  the  subdominant,  and  then  that  on  the  dominant  (35,88  f.;cf.  Prout,  p.  133 f.,  above). 
On  the  other  degrees  of  the  scale  it  is  rarely  used.  Thus  one  might  even  claim  that,  far 
from  being  disturbed,  the  chord  is  perhaps  rather  made  tolerable  by  the  influence  of  a 
distinctly  tonic  reference.  This  would  be  confirmed  by  the  fact  that  the  peculiar  character 
of  the  six-four  chord  was  recognised  long  before  tonality  had  come  clearly  to  the  surface 
of  the  musical  consciousness.  In  fact,  its  character  was  then  more  stringently  unique 
than  later. 

w.  F.  M.  1- 


178  THE  FACTORS  THAT  MODIFY  PARAPHONY  [ch. 

not  only  must  the  rhythmical  exposure^  be  avoided,  but  a  cogent  melodic 
line  must  also  be  driven  through  the  bass  note.  And  at  least  a  great 
part  of  the  necessity  for  also  approaching  a  bass  fourth  in  a  special 
way  is  likewise  due  to  this  proximity  of  the  major  third.  If  the  question 
be  raised  how  this  third  can  possibly  disturb  the  melodic  flow  of  the 
bass  when  even  the  fourth  itself  is  not  yet  sounded,  we  must  answer 
by  drawing  attention  again  to  the  fact  that  a  melodic  point  that  is 
about  to  follow  upon  one  just  sounded,  provides  a  sort  of  outlet  for 
the  latter's  residual  energy.  If  the  second  point  lies  near  the  first,  the 
discharge  is  easy  and  cogent.  The  further  apart  the  two  points  lie,  the 
less  forceful  is  the  transference.  A  melodic  leap  is  in  itself  a  difficulty. 
But  this  difficulty  can  be  increased  by  various  circumstances  :  by  the 
unfamiliarity  of  the  leap  to  be  taken,  and  especially  by  the  proximity 
of  an  easier — more  consonant,  more  paraphonic,  or  more  familiar — 
interval.  The  difficulty  of  the  fourth  from  the  bass  is  probably  due  to 
this  latter  circumstance.  The  fourth  is  more  consonant,  it  is  true; 
but  the  (major)  third — as  the  most  consonant  (i.e.  best  fused)  paraphony 
— is  both  very  important  and  very  interesting.  And  the  intonation  of 
a  bass  fourth  brings  any  slender  (leaping)  melodic  line  so  close  to  this 
attractive  third  that  the  actual  fourth  will  come  as  a  sort  of  jar  upon 
the  expectation.  The  melodic  flow  will  be  disturbed.  Hence  the  necessity 
of  making  the  melodic  line  leading  to  the  bass  note  cogent  by  the  devices 
summarised  above  (p.  134  f.). 

When  the  bass  note  is  firmly  established,  the  third  suggested  is 
that  upon  it.  When  it  is  not  well  established,  the  third  suggested  must 
be  that  below  the  upper  note  of  the  fourth.  This  is  strongly  indicated 
by  the  effort  all  the  rules  make  to  improve  the  cogency  of  the  bass 
melody.  As  the  bass  is  the  most  exposed  voice  of  all,  any  vacillation 
in  its  course  will  make  the  whole  chord  unstable. 

But  there  can  be  no  such  instability  when  the  fourth  is  not  on  the 
bass.    When,  as  in  cegc^,  it  lies  in  the  upper  voices,  all  the  rest  of  the 

^  Rhythmical  exposure  is  of  the  greatest  importance  in  music  generally  in  so  far  as 
it  increases  or  decreases  the  good  or  bad,  desired  or  undesired,  feature  of  any  purely  tonal 
element.  The  arrestive  effect  of  the  last  chord  of  a  cadence,  for  example,  is  heightened 
by  such  exposure,  as  it  must  be  if  the  'cadential'  effect  is  to  be  attained.  It  is  at  such 
points  that  one  sees  the  continuity  between  the  'dynamics'  of  tone  and  of  rhythm — a 
large  subject  which  deserves  special  treatment.  But  that  can  only  be  given  when  we 
have  brought  the  dynamics  of  tone  into  acceptable  order.  The  djmamics  of  rhythm  are 
naturally  more  obvious  and  apparent.  Among  the  ancient  Greeks  the  dynamic  of  arrest 
was  induced  largely  by  lowness  of  pitch.  But  the  octave  also  appeared  as  a  concluding 
symphony  (16,  Index  sub.  'Symphone,'  'Grave,'  and  'Aigu').  Both  of  these  effects  are 
still  generally  valid. 


XX]  THE  FACTORS  THAT  MODIFY  PAR  APHONY  179 

chord,  including  its  bass,  calls^  for  c^,  which  is  at  the  same  line  a  great 
consonance,  whereas  b  would  give  a  great  dissonance.  Similarly  in  egc^ 
the  two  paraphonies  exposed  upon  the  bass  make  c^  easy,  while  b  would 
itself  as  leading  note  suggest  the  c^  above  it. 

This  course  of  reasoning  strongly  suggests  that  the  basis  of  the  bass 
fourth  anomaly  must  be  merely  one  of  a  class  of  similar  cases.  Now 
the  anomaly  of  the  fourth  is  due  to  its  tendency  towards  the  neighbouring 
third.  Hence  we  might  express  the  principle  of  a  class  of  such  cases 
as  due  to  the  proximity  to  one  another  in  size  of  intervals  of  different 
fusional  or  paraphonic  character,  and  to  a  resulting  tendency  of  the 
one  to  point  to,  or  to  fall  into,  the  other. 

Methods  of  'ear-training'  make  use  of  this  feature  of  intervals  as 
well  as  of  others.  Thus  the  scheme  propounded  by  Bridge  and  Sawyer 
(3,  194)  is  as  follows  :  Octave — perfect  unison  of  the  two  sounds  so  that 
they  sound  almost  as  one;  VII — upper  note  will  require  to  ascend; 
7 — both  notes  will  require  to  move,  upper  down  one  degree,  lower  up 
a  fourth;  VI — upper  will  tend  to  fall  to  the  dominant;  6 — very  strong 
tendency  to  the  dominant;  5 — sounds  bright,  ear  at  rest;  diminished 
5 — both  notes  will  require  to  move  towards  one  another  by  a  second; 
augmented  4 — both  will  require  to  move  away  from  one  another  by  a 
second;  4 — upper  requires  to  fall  to  the  mediant;  III — calm,  peaceful 
effect,  ear  at  rest;  3 — peaceful,  but  more  melancholy,  ear  at  rest; 
II — close  discord,  lower  note  requires  to  descend  a  semitone;  2 — a 
fierce  discord,  lower  note  requires  to  descend  a  whole  tone. 

These  rules  for  the  recognition  of  intervals  make  use  of  a  variety 
of  aspects  :  (a)  symphony — octave  and  fifth ;  (6)  paraphonic  excellence — 
III,  3;  (c)  generalised  'harmonic'  (i.e.  polyphonic  or  paraphonic)  dis- 
position— 7,  diminished  5,  augmented  4,  II,  and  2 ;  (d)  tendency  towards 
proximal  interval  of  highly  symphonic  or  paraphonic  nature — VII  to 
octave,  VI  and,  still  more,  6  to  fifth,  4  to  major  third.  Other  distributions 
of  these  aspects  seem  to  be  possible,  for  example  :  (a)  symphony  in  its 
grade — 0,  5,  4;  (6)  paraphonic  excellence  or  repose — III  (small  and 
bright),  3  (small  and  sad),  VI  (large  and  bright),  6  (large  and  sad,  or — ^as 
discord — tending  to  fall  to  fifth);  (c)  dissonance  and  size — smallest  2, 
next  II,  largest  VII,  next  7,  middle  size  the  augmented  4  and  diminished 
5;  (d)  proximity  to  a  marked  consonance — VII  to  octave,  2  to  unison; 
second  degree  of  proximity — 7  and  II;  (e)  generalised  'harmonic' 
tendency — as  familiar.  Or  other  rules  might  be  formulated  according 
to  the  taste  and  fancy  of  the  recogniser. 

12—2 


180  THE  FACTORS  THAT  MODIFY  PARAPHONY  [ch. 

Similarly  Bridge  and  Sawyer's  rules  for  the  recognition  of  single 
■tones,  which  presuppose  a  given  (and  retained)  tonic,  are  :  tonic  (and 
octave) — at  rest;  leading  tone — up;  submediant — to  the  dominant; 
dominant — bright,  bare,  satisfying  to  the  ear;  subdominant — dull,  to 
the  mediant;  mediant — a  calm,  peaceful  sound,  on  which  the  ear  may 
rest;  supertonic — to  the  tonic. 

We  do  not  feel  any  surprise  at  the  major  seventh's  reminding  us 
of  the  octave  nor  of  the  minor  second's  suggestion  of  unison.  In  each 
case  the  other  interval  is  nearly  there,  as  far  as  the  volumic  proportions 
are  concerned,  and  is  a  very  characteristic  thing.  Apart  from  the 
tendency  of  the  fourth  to  the  major  third,  what  is  most  notable  is  the 
connexion  established  between  the  major  sixth  and,  in  a  stronger 
degree,  between  the  minor  sixth,  and  the  fifth.  We  might  have  expected 
both  these  intervals  to  share  the  repose  characteristic  of  the  thirds. 
The  major  sixth  undoubtedly  does  so  markedly;  and  it  is  debatable 
whether  it  does  not  really  do  so  much  more  than  it  points  to  the  fifth. 
But  in  the  minor  sixth  this  latter  tendency  is  indisputable.  Here,  then, 
we  have  another  example  to  the  fourth — a  paraphonic-consonant 
interval  with  a  tendency  of  'resolution.' 

In  fact,  we  have  more — an  exaggeration  of  the  feature  of  dissonance 
often  ascribed  to  the  fourth  :  for  the  minor  sixth — at  least  in  equal 
temperament — is  an  undoubted  dissonance  whenever  it  functions  as 
an  augmented  fifth.  It  is  so  even  when  the  chord  it  appears  in  otherwise 
contains  no  dissonance  at  all — ceg$  (two  major  thirds).  When  we 
substitute  e\}  or/ for  e,  we  get  (ce^al?)  the  first  inversion  of  the  common 
chord  on  the  major  tonic  a [7  {=  gtin  equal  temperament)  or  th^  second 
inversion  of  the  minor  common  chord  on/  {cfa\}).  Both  of  these  chords 
are  generally  consonant,  apart  from  the  bass  fourth  in  the  latter.  In 
particular  the  minor  sixth  is  now  consonant,  although  in  the  six-four 
chord  it  still  shows  a  tendency  to  fall  to  the  g  (the  'dominant,'  as  it 
were,  of  itself  and  of  the  rules  quoted  above). 

Of  course  the  problem  of  the  dissonance  of  c{e)gt'  is  not  solved  by 
a  simple  reference  to  the  beating  of  partials.  For  although  the  third 
partial  of  c  (=  g^)  and  the  second  partial  of  g^  {=  g^t)  beat  with  one 
another — upon  a  c  of  144  vibrations  per  second — 18  times  a  second, 
if  these  partials  occur  in  the  given  primary  tones,  we  must  not  forget 
that  the  fourth  partial  of  c(=  c^)  and  the  third  partial  of  e(=  b^)  beat 
36  times  a  second,  or,  if  the  interval  is  lowered  an  octave,  18  times  a 
second.  If  the  major  third  is  not  a  dissonance  on  144  vibrations,  it 
cannot  be  a  consonance  on  72  vbs.  if  cg^  is  a  dissonance  on  144  vibrations 


XX]  THE  FACTORS  THAT  MODIFY  PARAPHONY  181 

because  of  its  18  beats  per  second.  The  same  argument  applies  to  the 
interval  ea\f,  which  would  give  27  beats  a  second.  We  must,  therefore, 
look  for  some  other  difference  than  that  of  a  few  beats  per  second.' 
After  all,  the  dissonance  of  a  major  second  has  only  some  25  beats 
(between  the  primary  tones)  in  this  region,  so  that  it  should  sound 
better  than  the  minor  sixth  just  mentioned,  unless  we  admit  that  the 
beats  of  partials  are  of  less  effect  than  those  of  primaries.  But  why 
should  they  be  so?  And  if  they  are,  and  if  cgt  has  no  partials,  or  at 
least  not  those  that  make  18  beats  per  second  (only  one  of  the  beating 
partials  need  be  absent!),  then  cg^  would  be  consonant.  But  we  cannot 
revive  the  problem  of  partials  as  a  whole  again  here. 

In  both  the  chords  cefjaj?  and  c/aj?  we  have  a  grouping  of  a  minor 
third,  a  fourth,  and  a  minor  sixth.  They  differ  only  in  their  distribution. 
In  ce^  we  have  two  major  thirds  and  (in  equal  temperament)  a  minor 
sixth.  If  we  argue  that  in  the  latter  chord  the  outer  interval  is  heard 
as  an  augmented  fifth,  we  must  ask  :  why?  It  is  certainly  often  claimed 
that  the  ear  has  the  wonderful  faculty  of  hearing  what  a  chord  should 
be  ideally  instead  of  what  it  really  is.  Equal  temperament  is  commonly 
said  to  rest  upon  this  basis.  But  how  does  the  ear  know  what  should  be? 
Why  does  the  displacement  of  the  middle  tone  by  a  semitone  upwards 
or  downwards  incline  it  to  hear  cgrj  (=  ab)  rather  as  a  minor  sixth  and 
consonant  than  as  an  augmented  fifth  and  dissonant?  Surely  not  the 
pitch  of  the  middle  tone,  but  the  intervals  it  forms  with  the  outer 
tones.  Then  a  minor  third  and  a  fourth  somehow  call  for  a  minor 
sixth,  while  two  major  thirds  call  for  an  augmented  fifth.  But  why 
should  not  two  major  thirds  in  the  first  instance  have  called  for  (^# 
to  be  a  consonance  as  well  as  cajj?  If  answer  be  made  that  g^  is  not  a 
tone  of  the  original  scale,  why,  we  must  ask,  did  not  this  relation  of  it 
to  two  major  thirds  call  for  its  introduction  into  the  scale,  as  other 
chordal  connexions  have  called  for  changes  in  scales  (e.g.  the  sharp 
leading  tone  in  the  minor  scale)  ? 

The  question  is  certainly  difficult.  It  would  be  foolish  to  try  to  make 
it  look  easy.  Probably  the  best  reason  is  the  one  that  appears  first  in 
the  musical  consciousness  :  that  ceg^,  because  of  its  major  third  in  the 
bass  and  the  approximation  of  the  two  other  intervals  to  the  corre- 
sponding ones  of  the  major  triad  reminds  us  so  strongly  of  this  triad 
that  we  feel  the  discrepancy  as  a  distinct  jar,  confusion,  or  dissonance, 
just  as  we  may  feel  a  certain  approximation  to  the  portrayal  of  a  familiar 
face  as  a  caricature  or  as  an  artistic  offence.  This  feeling  can  only  be 
supported  by  the  fact  that  the  interval  in  question — even  as  a  minor 


182  THE  FACTORS  THAT  MODIFY  PARAPHONY  [ch. 

sixth — stands  very  low  in  the  scale  of  fusions.  When  the  minor  sixth 
stands  alone,  mere  proximity  to  the  fifth  would  make  it  suggest  the 
latter  very  strongly  and  so  appear  dissonant.  Its  paraphonic  capacity 
would  not  yet  have  been  exposed  by  its  subdivision  into  a  minor  third 
and  a  fourth.  So  we  find  a  prevailing  tendency  in  early  music  to  class 
the  minor  sixth  as  a  dissonance  (cf.  60,  2). 

Thus  we  come  upon  the  general  problem  of  the  natural  tendencies 
of  intervals  to  suggest  one  another.  Of  this  the  problem  of  the  resolution 
of  dissonances  forms  a  sub-division. 

Probably  the  chief  consideration  in  the  latter  problem  must  be  the 
cogency  of  the  melodic  movement  of  the  voices^  that  constitute  the 
dissonant  interval — an  interval  that  impels  melodic  movement  forwards 
and  so  caUs  for  'resolution.'  Thus,  for  example,  the  tritone  is  commonly 
said  to  resolve  generally  as  a  diminished  fifth  inwards  to  a  major  third 
and  as  an  augmented  fourth  outwards  to  a  minor  sixth.  If,  on  the 
basis  of  this  generalisation,  we  were  to  claim,  or  to  attempt  to  establish 
by  experimental  means,  some  special  connexion  between  these  intervals, 
we  should  not  only  find  ourselves  greatly  embarrassed  by  the  existence 
of  other  forms  of  resolution  of  the  tritone,  but  we  should  find  it  hard 
to  justify  the  distinction  between  the  two  forms  of  the  interval — 
diminished  fifth  and  augmented  fourth, — and  their  so  different  resolu- 
tions. A  much  easier  and  more  natural  solution  would  be  to  claim  that 
the  tritone  must  generally  'resolve'  by  the  most  cogent  motion  of 
the  voices  it  carries  on ;  and,  apart  from  other  modifying  circumstances, 
the  smallest  step  is  the  most  cogent.  Hence  the  best  resolution  is  by  the 
movement  of  a  semitone  in  each  voice  (cf,  above,  p,  172,  note).  And,  unless 
we  are  to  leave  the  key  of  the  chord  to  be  resolved,  this  procedure  would 
give  the  commonest  resolutions  of  the  two  forms  of  the  tritone  mentioned. 
We  should  by  no  means  thereby  be  precluded  from  making  other 
resolutions  of  a  chord  containing  the  tritone,  such  as  by  movement 
of  only  the  lower  voice  of  the  tritone  by  a  semitone,  or  by  different 
movements  of  the  other  elements  of  the  chord  than  those  of  the  tritone. 
We  should  expect  the  'harmonic'  value  of  all  possible  resolutions, 
i,e,  their  frequency  in  the  great  masters  or  their  beauty,  to  vary  directly 
with  the  melodic  cogency  of  the  different  voices  of  the  chord,  not 

*  Cf,  D,  F.  Tovey's  interesting  remark:  "Even  the  modem  researches  of  Helmholtz 
fail  to  represent  classical  and  modem  harmony,  in  so  far  as  the  phenomena  of  beats  are 
quite  independent  of  the  contrapuntal  nature  of  concord  and  discord,  which  depends  upon 
the  melodic  inteUigibihty  of  the  motion  of  the  parts"  (74), 


xx] 


THE  FACTORS  THAT  MODIFY  PARAPHONY 


183 


forgetting,  of  course,  that  the  cogency  of  a  movement,  though  primarily 
determined  by  its  shortness,  may  be  increased  by  other  factors,  such 
as  consonance  (step  of  fourth,  fifth,  or  octave),  tonaUty  (as  in  the  leading 
note),  an  established  figure  or  'sequence,'  the  symphonic  coherence  of 
one  or  even  two  otherwise  undetermined  movements  with  the  specially 
motivated  and  cogent  movements  of  the  other  voices,  etc. 

The  range  of  ordinary  variations  may  be  exemplified  more  fully 
with  the  interval  of  the  minor  seventh  on  the  dominant  in  the  keys  of 
c  major  and  minor.  It  is  obviously  not  here  a  case  of  the  mutual 
suggestiveness  of  intervals,  but  of  the  melodic  movements  of  voices 
resulting  in  different  intervals  which  are  then  accepted  in  so  far  as 
they  are  consonant  or  paraphonic  or  conform  with  the  intended  tonality 
or  are  possible  companions  for  the  tones  resulting  from  the  other  voices, 
and  so  on. 


i 


w 


EE 


Nor  do  we  need  to  assume  the  existence  of  a  mutual  suggestiveness 
between  whole  chords  as  such  (or  between  'harmonies').  Any  such 
connexions  or  suggestions  that  may  occur  and  be  felt  are  merely  a 
scheme  or  shorthand  of  melodic  connexions,  having  as  a  scheme  no 
binding  force  or  compulsoriness  at  all.  That  belongs  essentially  to  the 
melodic  transitions  in  detail  and  in  their  mutual  compatibihty,  as 
already  indicated.  Thus  in  the  generalised  connexion  between  the 
chord  of  the  dominant  seventh  and  that  of  the  tonic  there  is  clearly 
a  parallelism  of  the  patterns  that  result  from  the  combined  melodic 
transitions  or  resolution.  Neighbouring  tones  of  each  chord  lifted  up 
an  octave  retain  their  melodic  connexions  unchanged. 


In  the  case  of  the  connexion  of  chords  by  tonality,  we  reach  a  new 
level  of  melodic  grammar,  so  to  speak.  Instead  of  constructing  from 
single  words  we  make  use  of  familiar  phrases.  The  connexion  between 
dominant  or  subdorainant  and  tonic  or  vice  versa  is  so  familiar  and 


184         THE  FACTORS  THAT  MODIFY  PARAPHONY     [ch.  xx 

their  chords  so  characteristic  that  we  come  to  know  exactly  what  to 
expect.  Thus  the  melodic  connexion  establishes  itself  more  easily. 
Besides, — and  perhaps  this  is  the  root  of  the  matter, — in  these  progres- 
sions the  very  symphonic  effect  produced  may  be  just  what  is  desired 
at  that  point  of  the  music,  namely,  a  tendency  towards  arrest  or  clo^e. 
The  existence  of  an  effect  due  to  the  connexion  of  chords  a  third 
apart  has  been  claimed  by  Shinn,  who  believes  in  a  grading  of  effect 
in  three  steps,  corresponding  to  fifth-fourth,  sixth-third  and  seventh- 
second  intervals  between  the  chords.  To  some  extent  Shinn  fails  to 
distinguish  these  connexions  from  the  intervals  between  the  pairs  of 
voices  in  question.  The  establishment  of  this  grading  would  require 
an  extended  statistical  foundation,  chosen  with  due  regard  to  the 
elimination  of  other  differences.  The  problem  raised  by  Shinn  is  of 
considerable  importance,  but  the  factual  and  deductive  grounds  for 
his  generalisation  seem  to  be  at  present  insufficient. 

It  has  been  claimed  occasionally  that  downward  motion  favours 
melodic  continuity,  and  so  tends  to  annul  unparaphonic  effects.  Here 
again  we  lack  a  proper  basis  from  which  to  confirm  or  to  oppose  this 
suggestion.  We  know  that  descending  intervals  are  found  harder  to 
judge  at  first  than  are  ascending  ones.  Whether  the  intervals  of  melodies 
descend  oftener  than  they  ascend  has  yet  to  be  determined  (cf.  16,  173  ff.). 
The  effect  of  a  progression  of  roots  of  chords  by  leap  of  a  third  is  said 
to  be  generally  more  satisfactory  downwards  than  upwards,  especially 
if  two  or  three  such  leaps  occur  in  succession  (52,  56;  cf.  60,  35). 


CHAPTER  XXI 

RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY 

The  division  of  intervals  into  the  three  classes  of  symphony,  paraphony, 
and  diaphony,  which  we  have  attained  and  expounded  in  the  previous 
chapters,  stands  in  opposition  to  the  progressive  grading  of  intervals 
derived  from  the  studies  of  Helmholtz  and  Stumpf.  In  these  the  octave 
forms  an  extreme  fusion  of  apparent  unity  in  simultaneity  or  smoothness 
in  transition  between  tones,  from  which  there  is  a  gradual  variation 
towards  complete  two-ness  or  harsh  beating  in  simultaneity  or  complete 
difference  in  succession.  We  have  reflected  briefly  from  time  to  time 
upon  this  change  of  theory,  but  it  seems  necessary  to  consider  it  more 
explicitly  now. 

Helmholtz's  quantitative  estimation  of  grades  of  consonance  between 
simultaneous  tones  was  based  upon  mathematical  calculations  regarding 
the  roughness  of  beating  between  their  component  partials  (20, 192  flf.). 
In  so  far  as  a  connexion  between  successive  tones  might  be  established 
by  their  common  partials  a  certain  smoothness  of  transition  could  be 
achieved  without  which  tones  would  be  simply  different;  but  there 
could  be  no  harshness  or  dissonance.  In  the  case  of  simultaneous  tones 
beating  would  also  appear  so  far  as  the  partials  presupposed  actually 
existed  in  the  tones;  and  the  roughness  felt  in  any  interval  would 
vary  with  the  partials  contained  in  its  tones.  But  in  so  far  as  the 
partials  might  lapse  ^^'ithout  alteration  of  the  grading  of  intervals 
according  to  consonance  and  dissonance,  the  theory  would  be  fatally 
discredited.  No  force  of  memory  could  save  it.  It  is  necessary  to  recall 
these  points  from  a  previous  chapter  because  they  compel  us  to  face 
the  task  of  constructing  a  theory  of  consonance  and  dissonance  without 
the  help  of  coincident  or  beating  partials,  except  it  may  be  as  a  secondary 
or  supporting  factor  making  for  confusion  or  loss  of  distinction. 

Stumpf's  grading  is  based  upon  direct  experimental  observations 
confirmed  and  amplified  by  the  work  of  others.  His  description  of  fusion, 
especially  in  its  highest  grades,  as  approximation  to  the  unity  of  a 
single  tone,  is  a  true  representation  of  the  experimental  judgments 
and  corresponds  fully  with  the  descriptive  definition  of  'symphony' 
given  by  the  Greeks,  which  is  thus  seen  to  be  correct  both  as  far  as  it 
goes  and  inasmuch  as  it  does  not  venture  farther.    We  have  adopted 


186     RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY     [ch. 

it  in  our  exposition  above,  and  have  found  a  sufficient  and  satisfactory 
parallel  to  it  in  the  volumic  coincidence  of  tones,  at  least  for  the  cases 
ef  the  octave  and  the  fifth.  These  yield  a  volumic  mass  that  in  two 
distinct  grades  resembles  the  symmetrical  balanced  mass  of  sound 
that  constitutes  the  pure  tone.  There  is  no  doubt  that  this  part  of  the 
theory  must  be  maintained  in  all  future  theories  of  consonance  and 
dissonance.  It  fully  explains  the  symphony  of  the  octave  and  the  fifth. 
But  in  the  other  two  divisions  of  paraphonic  and  diaphonic  intervals 
our  scheme  fails  to  coincide  with  Stumpf's.  It  is  only  in  so  far  as  we 
can  look  upon  intervals  as  mere  static  and  isolated  masses  of  sound 
that  we  can  attempt  to  force  the  aspect  of  'approximation  to  unity' 
down  the  whole  series  from  octave  to  major  seventh.  But  this  static 
aspect  is  practically  excluded  from  music  and  is  hard  to  realise  experi- 
mentally unless  with  unmusical  persons  or  with  neglect  of  analysis 
(cf.  Kemp,  29)^.  The  new  description  is  one  that  establishes  a  plain 
continuity  between  intervals  (music  in  two  voices)  and  music  in  any 
number  of  parts.  In  paraphony,  we  have  shown,  there  is  an  equal 
distinguishability  of  the  tones  of  an  interval,  so  that  two  melodies  may 
run  side  by  side  through  it  without  any  mutual  interference.  In  diaphony 
this  confusion  re-appears,  although  the  tones  are  not  now  apparently 
one,  but,  as  we  can  readily  infer  from  their  harsh  coincidence,  obviously 
more-than-one-like.  The  major  seventh  and  the  minor  second  strongly 
declare  their  dissonance.  But  they  do  not  necessarily  Umit  themselves 
to  a  semblance  of  two-ne&s,  as  the  erroneous  judgments  of  untrained 
and  unmusical  ears  often  indicate  (64,  37iff.,ei  fassim).  We  have  to 
find  a  sufficient  basis  for  this  positive  diaphony  (or  dissonance,  which 
in  music  is  always  felt  as  a  positive  harshness  not  sufficiently  described 
by  mere  two-ness  or  difference^)  in  the  stuff  of  tones,  and — to  emphasise 
it  again — without  the  help  of  beating  partials.  Even  beating  difference- 
tones  must  be  kept  in  reserve  for  any  special  purpose  they  may  properly 
be  able  to  fulfiP. 

*  Kemp  and  Kulpe  go  much  farther  than  Stumpf,  who  refuses  to  see  any  change 
in  a  fusion  owing  to  the  presence  of  other  tones. 

^  A  much  better  description  is,  for  example,  this:  " discordantia  est  duorum  sonorum 
sibimet  permixtorum  dura  collisio"  {Quiadam  Aristotdes,  6,  26o). 

*  A  claim  to  the  recognition  of  neutral  grades  of  sonance  was  made  by  F.  Krueger 
in  his  lengthy  discussion  of  Stumpf's  criticism  of  his  difference-tone  theory  of  consonance 
and  dissonance.  Krueger  pointed  out  that  the  psychological  relativity  of  the  notions 
of  consonance  and  dissonance  and  the  historical  mobility  of  the  border  between  them 
was  noted  by  Helmholtz.  We  have  already  seen  how  little  real  truth  there  is  in  this 
historical  appeal.  Stumpf  was  also  tempted  to  form  a  class  of  neutral  intervals, — the 
sevens  in  particular, — but  was  clearly  aware  that  his  notion  of  fusion  could  give  no  such 


XXI]    RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY     187 

Now  the  whole  course  of  our  new  theory  of  tones  advises  us  to  look 
around  the  main  points  of  our  actual  conclusions  for  lines  of  develop- 
ment rather  than  away  from  them  to  outlying  regions  or  to  adventitious 
phenomena.  Our  grading  gives  three  regions  with  special  points  in 
each,  namely  : 

Symphony.  Paraphony,  Diaphony. 

Loss  of  distinction  in  Ease  and  equality  Loss  of  distinction  in 

unitariness  of  distinction  confusion 

p,  o,  5,  4  III,  3,  VI,  6  T,  7,  II,  2,  VII 

(We  have  added  the  prime  to  the  head  of  the  series  as  our  study  of  its 
functions  in  music  has  justified  our  doing  so.  The  prime,  as  the  older 
writers  on  music  so  often  said^,  and  as  the  modern  (acoustical)  theory 
of  music  has  always  failed  to  understand  or  to  explain 2,  is  the  greatest 
consonance  of  all.) 

opposition  &s  is  implicit  in  the  concepts  of  consonance  and  dissonance.  Krueger's  theory 
does  not  really  justify  his  formation  of  a  neutral  class  either,  however  true  it  may  be  that 
various  intervals  might  well  be  placed  in  either  class  of  consonance  or  dissonance.  And  it 
is  difficult  to  see  how  such  an  appeal  to  a  neutral  class  can  save  a  theory  from  special 
criticism  after  it  has  been  shown  that  such  very  different  theorists  as  Helmholtz  and  Stumpf 
also  noted  the  existence  of  intervals  of  doubtful  or  indifferent  character  and  yet  equally 
failed  to  justify  the  formation  of  a  neutral  class.  Krueger  beheved  that  an  interval  tended 
towards  neutrality  of  sonance  in  so  far  as  it  was  sounded  more  gently  or  briefly,  or  was 
raised  towards  the  higher  end  of  the  musical  range  of  pitch,  or  was  extended  by  one  or 
more  octaves,  etc.  {v.  31,  passim).  Neutrality  is  given  when  "for  any  reasons  the  con- 
sonantal or  dissonantal  characteristics  postulated  by  the  theory  are  lacking  or  reach  a 
certain  limit  of  clearness  or  of  effectiveness  in  the  whole  soimd,  or,  again,  when  one  and 
the  same  chord  contains  both  consonantal  and  dissonantal  characteristics,  and  these 
two  sets,  not  greatly  marked,  annul  one  another  in  the  total  impression"  (32,  su). 
Consonance  is  due  to  the  coincidence  of  difference-tones,  of  which  Krueger  asserts  the 
existence  of  five  or  six  (theoretically)  for  each  interval;  dissonance  to  the  'neighbourly 
interference '  of  two  or  more  of  these  five  or  six,  i.e.  to  their  beating  and  indistinguishability, 
etc.  Thus  we  get  an  approximation  towards  unity  and  indistinguishability  both  in  the 
greater  dissonances  and  in  the  greater  consonances.  Why  two  primary  tones  and  two 
loud  difference-tones  should  suggest  more  oneness  than  these  two  primaries  with  five 
difference-tones  that  are  proportionately  weaker  is  not  explained:  especially  when  we 
recall  that  the  difference-tones  need  not,  and  in  the  vast  majority  of  cases  are  not  dis- 
tinguished at  all.  Surely  the  fewer  more  distinct  elements  should  be  'rougher'  than  the 
fainter  ones.  Besides,  Stumpfs  criticism  of  the  facts  regarding  these  five  or  six  difference - 
tones  robs  the  theory  of  its  basis  (cf.  68,  69  and  70). 

^  E.g.  Johannes  de  Garlandia  (6,  104):  "  Concordantia  dicitur  esse  quando  due  voces 
junguntur  in  eodem  tempore,  ita  quod  ima  potest  compati  cum  alia,  secundum  auditum... 
Perfecta  dicitur  quando  due  voces  jungimtur  in  eodem  tempore,  ita  quod  una,  secundum 
auditum,  non  percipitur  ab  alia  propter  concordantiam,  et  dicitur  equisonantiam,  ut  in 
unisono  et  diapason."  Unison,  a«  already  said,  has  only  meaning  for  polyphony;  for 
only  then  can  one  sound  be  held  and  heard  to  be  the  conjunction  of  two. 

*  Cf.  64,  i7«:  "the  hypothetical  fusion  of  the  prime." 


188     RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY    [ch. 

In  order  that  we  may  look  around  these  intervals  for  some  means 
of  progress,  let  us  arrange  them  according  to  their  place  in  the  ordinal 
series  of  an  octave.   Then  we  get : 


nph. 

Diaph. 

Paraph. 

Diaph. 

Sjmiph. 

Diaph. 

P 

2,  II 

3,  III 

? 

4 

T 

0 

VII,  7 

VI,  6 

6 

5 

T 

The  two  ends  of  the  series  are  constituted  by  the  greatest  symphonies; 
next  them  stand  the  greatest  diaphonies ;  a  lesser  diaphony  leads  through 
the  two  grades  of  paraphony  to  a  middle  region  in  which  stand  the 
consonances  of  the  fourth  and  fifth,  separated  by  the  diaphonic  tritone. 
This  arrangement  suggests  a  regular  or  periodic  change  from  the  extremes 
towards  the  middle  region,  in  which  range  the  diatonic  intervals  mark 
out  familiar  points. 

But,  we  must  ask,  do  these  intervals  really  designate  'points  in  the 
range  of  intervals  included  within  the  octave  ?  If  both  the  minor  and 
the  major  second  are  diaphonies,  and  both  the  minor  and  the  major 
third  are  paraphonies,  should  we  not  rather  speak  in  general  of  a 
diaphonic  or  paraphonic  regioni  And  should  we  not  also  in  particular 
speak  of  a  region  having,  for  example,  the  paraphonic  degree  of  the 
minor  third?  (Remember  again  that  we  have  lost  the  treacherous  aid 
of  beating  partials !) 

Arguments  of  some  strength  may  be  urged  in  support  of  such  a 
view,  startling  and  even  outrageous  as  it  may  at  the  first  glance  seem 
to  be^.    These  rest  upon  facts  that  are  fully  familiar  to  students  of 

^  One  can  see,  however,  that  Stumpf  felt  himself  drawn  towards  this  conclusion 
(cf.  64,  176  fl.):  "Doubts  about  the  sinking  of  the  curve  between  [major  and  minor  thirds, 
etc.]  have  another  special  cause  in  that  these  intervals  differ  only  by  a  semitone  and  so 
the  intervening  are  always  apprehended  as  a  deviation  from  them,  as  a  mistuned  third 
etc."  So,  also,  Helmholtz,  who  wrote  (15,  200):  "two  simple  tones  making  various  intervals 
adjacent  to  the  major  third  and  sounded  together  will  produce  a  uniform  uninterrupted 
mass  of  sound,  without  any  break  in  their  harmoniousness,  provided  they  do  not  approach 
a  Second  too  closely  on  the  one  hand  or  a  Fourth  on  the  other.  My  own  experiments 
with  stopped  organ  pipes  justify  me  in  asserting  that  however  much  this  conclusion  is 
opposed  to  musical  dogmas,  it  is  borne  out  by  the  fact,  provided  that  really  simple  tones 
are  used  for  the  purpose."  The  later  experimental  work  of  Stumpf  and  Meyer  (60)  has 
disproved  this  in  so  far  as  the  ability  to  recognise  the  interval  with  great  approximation 
to  its  pure  form  is  concerned.  (Helmholtz  went  so  far  as  to  say  it  was  impossible  to  tune 
perfect  major  or  minor  thirds  on  stopped  organ  pipes  or  tuning-forks  without  the  aid  of 
other  intervals.)  Stumpf  and  Meyer's  results  prove  the  ability  to  find  the  almost  exact 
interval  even  with  successive  tones.  Here  there  is  presumably  no  'fusion'  of  any  kind, 
beats,  difference-tones,  or  'fusion'  proper, — so  that  we  are  all  the  more  thrown  back 
upon  an  accurate  sense  of  interval  (to  which  we  have  given  a  positive  basis  in  our  theory 
in  volumic  proportions,  and)  which  is  presumably  most  developed  in  the  most  highly 


XXI]    RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY      189 

hearing  and  of  music,  but  that  have  been  rather  coldly  entertained  by 
those  who  have  aspired  to  construct  a  theory  of  the  foundations  of 
music.  This  neglect  is,  of  course,  natural  and  inevitable  so  long  as  the 
facts  in  question  seem  to  be  not  only  incompatible  with  the  prevalent 
theoretical  point  of  view,  but  even  to  be  artificial  and  arbitrary  dis- 
tinctions or  classifications  unattainable  from  the  straight  roads  of 
theory. 

One  argument  is  familiar  to  all  those  who  possess  absolute  ear  and  to 
others  from  their  records.  Such  a  person  does  not,  for  example,  hear  the 
progressive  transition  from  a  good  d  to  a  good  e  as  a  sharp  lapse  oi  'd^ 
into  *  something  not  d,  but  higher  than  d/  then  '  (?#,'  then  '  flatter  than  e,' 
and  finally  'e,'  or  the  like;  but  as  a  region  of  'd  colour'  surrounding  the 
'  best  d,'  bordering  on  a  region  of  djj^  including  a  smaller  optimal  region, 
passing  into  a  new  region  of  e  leading  to  a  best  e,  and  so  on.  Each 
tone  name  characterises  not  one  rate  of  vibration  or  a  very  narrow 
zone  defined  by  the  range  of  hardly  distinguishable  differences,  but  a 
relatively  large  zone  of  perfectly  obvious  differences.  An  absolute  ear 
also  adjusts  itself  readily  enough  to  differences  in  the  pitch  to  which 
a  musical  instrument,  such  as  the  piano,  may  be  tuned,  if  the  difference 
is  not  great  enough  to  carry  a  tone  into  the  zone  of  its  chromatic  neigh- 
bour. In  this  case  keys  are  interchanged,  c  major  becoming  d\^  major 
or  b  major.  In  some  cases  of  very  rigid  absolute  ear,  however,  even 
smallish  differences  of  tuning  are  disturbing. 

Another  is  the  similar  usage  so  characteristic  of  music  of  applying 
the  same  pitch  names  to  zones  of  pitch.  Thus,  e.g.  we  have  not  only 
d\^,  d  and  <i#  but  also  d\^\^  and  dx  (##)^,  terms  which  are  acoustically 
so  anomalous,  as  they  are  on  instruments  of  equal  temperament  exactly, 
and  in  any  case  very  nearly,  equal  to  the  pitches  of  c  and  e.  Similarly 
the  names  of  the  intervals  are  not  only  the  same  for  their  major  and 
minor  forms,  but  they  are  extended  to  include  augmented  and  diminished 
forms.  An  augmented  second,  for  example,  is  more  or  less  the  same  as 
a  minor  third  as  far  as  pitch  is  concerned;  but,  as  music  has  always 
claimed,  the  augmented  second  partakes  of  the  diaphony  of  the  major 
second,  not  of  the  paraphony  of  the  minor  third.    And  an  augmented 

trained  observers  such  as  Stampf  and  Meyer  had.  Hence  in  observers  such  as  Helmholtz, 
who  do  not  possess  this  faculty,  we  may  assume  we  get  an  approach  to  the  eflFect  of  the 
consonantal  value  of  the  tones  apart  from  their  intervallic  prtn^ision  and  apart  from 
difiference-tone  values  which  Helmholtz  obviously  failed  to  make  use  of  in  these  cases. 
Hence  a  neutral  (or  as  I  call  it,  a  paraphonic)  range  from  second  to  fourth  or  thert^by. 

*  The  original  points  for  these  are,  of  course,  not  d,  but  d^  and  dZ  respectively,  so  that 
there  is  a  second  grade  of  flattening  or  8hari>ening  here  only  in  a  temiinological  sense. 


190     RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY    [ch. 

or  diminished  interval,  when  it  occurs  in  full  force,  as  between  a\}  and 
6  in  a  minor  scale  is  found  to  be  peculiarly  hard  to  sing.  When  the 
same  transition  of  pitch  occurs  in  a  true  minor  third,  as  from  c  to  e^ 
in  c  minor,  no  such  difficulty  is  felt. 

We  may  recall  a  quotation  from  Stumpf  (p.  144,  above)  :  "That  one 
and  the  same  unmodified  pair  of  tones  should  now  fuse  more  and  now 
less  according  as  we  apprehend  it  as  c-e\^  or  as  c-djf^  is  out  of  the  question, 
because  fusion  is  a  function  of  the  two  sensations — or  of  their  physio- 
logical bases — and  can  change  only  with  these  same"  (71,328),  This 
utterance  implies  the  very  sharp  distinctions  of  the  points  of  the  scale 
and  the  fusions  peculiar  to  them  that  we  have  already  mentioned. 
But,  if  pitches  and  degrees  of  paraphony  belong,  not  to  points  optimally 
and  only,  but  also  in  some  degree  to  zones  or  regions,  there  seems  to 
be  no  insuperable  barrier  to  our  extending  these  zones  for  various 
reasons  and  to  ascribing  to  a  tone  the  pitch  name,  or  to  an  interval 
the  paraphony,  of  the  zone  within  which  it  is  included.  Then  in  con- 
nexion with  these  musical  reasons  of  key  relationship,  modulation,  etc., 
we  shall  apprehend  a  tone  of  a  certain  number  of  vibrations  as  a  modified 
d,  i.e.  as  d^,  not  as  e\^,  with  which  it  may  more  or  less  coincide,  and  we 
shall  then  attach  to  it  the  pitch  name  of  the  (extended)  region  we  ascribe 
it  to  and  the  paraphony  of  that  region,  namely  the  diaphony  of  the 
seconds.  But  its  diaphony  will  not  be  greater  than  that  of  the  major 
second  as  is  the  minor  second's,  but  less,  of  course;  still,  however, 
diaphony,  and  not  the  paraphony  of  the  third.  The  paraphony  of  an 
interval  will  be  determined  primarily  and  chiefly  by  its  precise  nature 
as  an  interval,  but  within  any  zone  much  will  also  depend  upon  the 
circumstances  of  our  approach  to  it  in  the  flow  of  the  music. 

Reverting  to  our  series  within  the  octave  we  may  now  note  further 
that  the  great  symphonies  are  bounded  by  the  strongest  diaphonies. 
In  the  former  there  is  a  special  fit  or  balance  of  the  tones;  in  the  latter 
we  have  the  worst  coincidences  of  volume,  as  soon  as  the  range  of  hardly 
perceptible  difference  is  passed.  The  zone  of  symphony  in  the  octave 
and  fifth  is  very  narrow,  allowing  of  no  degrees,  no  major  or  minor 
forms.  In  the  diaphonies  bordering  on  the  symphonies  the  pitch  centres 
of  the  tones  either  encroach  upon  one  another  (small  intervals)  and  so 
make  each  other  hardly  distinguishable,  or  the  lower  end  of  the  volume 
of  the  upper  tone  and  the  pitch  centre  of  the  lower  fall  close  together 
(large  intervals),  making  a  notable  blemish  in  the  centre  of  the  lower 
tone,  and,  in  the  major  seventh,  an  approximation  to  the  octave  which 
it  thus  strongly  suggests.  We  probably  do  not  shift  our  point  of  attention 


XXI]    RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY     191 

in  these  two  groups  from  one  side  to  another  of  the  pitch  of  the  lower 
tone.  We  hear  each  tone  in  a  mass  as  a  whole,  including  its  pitch  and 
the  length  of  its  volume;  then  its  paraphonic  nature  will  be  decided 
by  whatever  excessive  proximity  of  parts  or  relative  freedom  and 
independence  of  them  there  is  on  the  whole. 

In  the  paraphonies  we  reach  a  zone  of  volumic  proportions  in  which 
the  pitches  of  the  tones  stand  well  apart,  are  therefore  easily  discriminable 
without  mutual  interference,  and  in  series  can  readily  form  distinct 
melodic  hues,  not  running  together  or  blurring  one  another. 

Round  the  fifth  again  we  find  a  border  of  diaphony  in  the  tritone 
and  in  the  minor  sixth  when  it  appears  as  an  augmented  fifth  or  when 
it  is,  as  in  early  music,  not  marked  off  by  its  place  in  chords  as  a  para- 
phonic sixth.  This  fluidity  of  character  confirms  the  extensibility  of 
zones  we  have  already  noted.  And  it  also  seems  a  sufficient  explanation 
of  the  fact  that  in  the  sixths  the  lesser  paraphony  follows  upon  the 
greater,  not  conversely  as  in  the  thirds.  If  the  fifth  stood  not  where 
we  find  it,  the  interval  next  beyond  the  major  sixth  (i.e.  the  minor 
sixth)  would  presumably  be  a  better  paraphony  than  it  actually  is. 

The  only  point  that  remains  for  discussion  concerns  the  fourth, 
that  old  bone  of  contention.  Our  scheme  tempts  us  anew  to  consider 
it  a  dissonance,  less  diaphonic  than  is  the  tritone.  So  one  might  plausibly 
account  for  its  having  been  so  often  termed  a  dissonance.  But  even  so 
it  might  at  the  same  time  be  held,  like  the  sixth,  to  be  includible  in  the 
paraphonic  zone  of  the  thirds,  being  then  a  little  more  paraphonic 
than  the  major  third.  But,  on  the  other  hand,  we  cannot  get  round 
the  fact  that  the  fourth  is  a  consonance  of  third,  though  low,  degree. 
To  what  extent  its  being  the  inversion  of  the  fifth  is  responsible  for  this, 
it  might  be  difficult  to  decide.  The  connexion  felt  between  its  symphony 
and  its  position  in  the  upper  voices  of  chords  would  support  any  doubt 
as  to  its  being  in  itself  symphonic,  as  would  also  the  large  (statistical) 
gap  between  the  fifth  and  the  fourth  and  the  small  gap  between  the 
latter  and  the  thirds  (cf.  77,  58).  Our  failure  to  find  a  form  of  distinctive 
volumic  balance  for  the  fourth  would  tend  towards  the  same  conclusion. 
So  we  should  infer  that  the  fourth  is  a  paraphony  or  even  a  diaphony 
whose  cousinship  to  the  fifth  has  given  it  the  rank  of  symphony.  This 
conglomeration  of  relations  would  suit  the  unstable  character  of  the 
fourth  in  music.  An  incautious  fourth  from  the  bass  would  then  be  a 
sort  of  augmented  third  or  a  diaphony  of  lesser  degree  than  the  tritone. 
It  would  have  the  fluid  nature  of  the  minor  sixth  between  diaphony 
and  paraphony,  sometimes  'resolving'  therefore  into  an  ordinary  (major) 


192      RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY    [ch. 

third.  There  is  in  the  relation  of  inversion  as  such  no  reason  for  any 
transference  of  symphony  from  the  fifth  to  the  fourth.  But  when  the 
fourth  occurs  as  the  top  of  a  major  common  chord,  it  is  then  the  comple- 
ment to  an  actual  fifth. 

But  if  the  range  of  the  octave  is  characterised  by  these  wide  regions 
of  paraphony  and  diaphony,  what,  the  reader  may  ask,  accounts  for 
the  very  precise  form  of  our  present  diatonic  scale  and  for  the  perfection 
that  so  many  claim  for  just  temperament.  Here  we  return  to  the  problem 
of  the  scale,  which  is  primarily  the  problem  of  the  skeleton  of  divisions 
of  the  octave  that  will  at  any  time  offer  the  greatest  and  most  convenient 
scheme  of  intervals  and  chords,  etc.  The  octave  and  fifth  are  inevitable 
land-marks  of  great  precision  and  force.  The  fifth  may  be  taken  upwards 
or  downwards,  yielding  cfgc^.  And  from  the  whole  tone  thus  derived 
we  may  get  various  farther  divisions  of  which  the  chief  type  attempts 
to  subdivide  the  two  large  intervals — the  two  'tetrachords.'  It  is  a 
familiar  fact  that  many  forms  of  subdivision  have  been  tried  and 
tolerated  as  more  or  less  permanent  modes.  We  ourselves  have  two — the 
major  and  minor,  the  latter  of  which  the  theories  that  have  worked 
with  harmonic  partials  have  never  been  able  to  justify.  And  in  bagpipe 
music  we  find  a  neutral  third  midway  between  our  major  and  minor 
sections.  The  extensibility  of  zones,  it  may  be  noted  in  passing,  is 
richly  confirmed  in  the  familiar  tendency  of  the  ear  trained  upon  our 
two  modes  to  hear  the  intervals  of  exotic  music  as  approximations  to 
the  intervals  from  the  same  zone  that  occur  in  our  music. 

Of  course  the  burning  question  is  :  why  has  our  harmonic  style  so 
favoured  our  two  sections  in  the  precise  form  they  have  taken,  and 
especially  the  major  one?  Here  the  appeal  to  the  correlation  with  the 
ratios  of  harmonies  seems  inevitable.  To  deny  may  seem  like  wilful 
scorn  of  a  providence  governing  musical  theory.  But  at  this  point 
difference-tones  seem  to  be  much  more  important  than  upper  partials. 
For  the  latter  are  not  only  accidental  and  distant,  but  as  Macfarren 
said,  they  should  really  create  quite  a  babel  of  confusion;  whereas  the 
loudest  difference  tones  of  4,  5,  and  6  would  necessarily  be  1  (twice), 
2  (twice),  3,  and  4;  or  with  the  primaries  1  (loud),  2  (loud),  3  (not 
weak),  4,  5,  and  6.  Here  we  have  two  octaves  1-2  and  2-4,  and  two 
fifths  2-3  and  4-6.  A  deviation  from  5  in  either  direction  would  upset 
the  two  octaves  and  one  of  the  fifths,  thus  creating  in  the  whole  sound 
a  somewhat  indefinitely  located,  but  quite  noticeable,  harshness.  The 
minor  third,  on  the  other  hand,  is  sufficiently  justified  by  its  being  the 
remainder  of  the  major  third  from  the  fifth  (which  two  would  between 


XXI]   RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY      193 

them  yield  all  the  tones  of  the  major  diatonic  scale).  Any  scale  deter- 
mined by  these  two  intervals  would  also  yield  minor  third  chords, 
which  would  thus  become  famihar  parts  of  the  system,  tolerated  a& 
such,  in  spite  of  their  less  symphonic  resultant,  which  is  familiar  to  every 
musician. 

Two  points  may  be  emphasised  with  reference  to  the  above.  First, 
we  are  not  here  attempting  to  justify  the  thirds  ab  origine  by  this  means; 
they  carry  their  own  justification  as  paraphonies  within  them;  the  only 
question  is  :  why  has  our  harmonic  music  so  favoured  the  particular 
points  of  the  paraphonic  zone  at  which  the  major  and  minor  thirds 
stand?  Second,  we  do  not  forget  that  the  major  mode  was  by  no  means 
the  fount  and  origin  of  all  music;  it  is  rather  the  culminating  result 
of  a  long  development.  But  it  stands  in  modern  music  as  the  logically 
prior  skeleton  of  our  system,  and  as  such  it  requires  special  justification. 

It  is  open  to  the  musician,  of  course,  to  make  many  other  sections 
of  the  octave  than  our  two,  and  to  build  a  perfectly  coherent  music 
of  somewhat  limited  scope  upon  them.  This  may  be  forcibly  extended 
even  some  way  into  the  range  of  polyphony.  But  the  fascination  of 
the  great  consonances  gradually  drags  it  into  the  Unes  of  the  two  chief 
steps  of  paraphony  and  the  great  systems  they  yield. 

With  this  free  outlook  we  seem  able  to  do  justice  to  all  kinds  of 
music  without  the  ridiculous  restrictions  and  sophistries  of  the  harmonic 
theories.  Of  course,  all  the  theories  try  to  bring  themselves  into  line 
with  music  so  as  to  explain  its  ways;  and  in  so  far  as  theories  succeed 
in  doing  so,  we  may  be  all  taken  as  in  agreement  with  one  another. 
The  only  question  is :  whose  basis  of  explanation  is  the  primary  one  ? 
And  we  may  claim  that  honour  for  ours  for  two  reasons  :  (1)  the  basis 
of  explanation  is  inherent  in  the  primary  tones  concerned  in  the 
phenomena  to  be  explained;  (2)  the  new  facts  we  have  built  upon  have 
not  been  derived  from  sources  external  to  music  so  as  now  to  be 
problematically  read  into  music,  but  they  have  been  derived  from  the 
empirical  generalisations  of  music  itself. 

The  explanation  suggested  for  the  nature  of  the  intervals  between 
the  fifth  and  the  octave  would  obviously  hold  equally  for  the  intervals 
greater  than  the  octave  up  to  the  twelfth.  For  in  the  ninths  the  ends 
of  the  upper  volume  will  fall  near  the  pitch  centre  of  the  lower  tone 
and  will  so  mar  its  outline,  while  in  the  tenths  and  eleventh  these 
points  will  draw  apart  so  that  they  will  be  readily  distinguishable  and 
independent.  In  the  twelfth  we  meet  again  (cf.  77,  68,  iiif.)  with  a 
sort  of  balance  or  symmetry  of  volumic  parts,  the  upper  half  of  the 

W.  F.  M.  13 


194    RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY     [ch. 


volume  of  the  lower  tone  being  divided  into  three  equal  parts  by  the 
pitch  and  the  lower  end  of  the  upper  tone's  volume.  So  we  may  well 
have  diaphonies  in  the  region  on  either  side  of  this  point.  But  there 
seems  to  be  no  further  ground  for  estabhshing  paraphonies,  such  as 
the  compound  sixths.  Perhaps  the  diaphonies  around  the  double 
octave  might  pass  as  blemishes  on  the  division  (4,  2,  1,  1)  peculiar  to 
the  latter. 

At  this  point,  feeling  we  had  pushed  the  range  of  the  direct  and 
inherent  distinctions  of  intervals  well  out  bevond  the  octave  without 


.5 

_!■_ 
I-l 

X 

x-2 

X 

x-3 


Relative   diminution   of  volume   in   higher   tone 


25     16 
24     13 


■10. 

10  9    8 

a  8   7 


.■30 
T 


D.  Z.  -  Diaphonic  Zone 


/*  Z.  =  Paraphonic  Zone 

Fig.  6. 


•40 
5 


6' -Symphony 


appealing  to  any  extraneous  or  secondary  factors,  we  might  perhaps 
venture  to  ask  the  functions  of  memory  to  support  us  or  invite  the 
aid  of  harmonics.  These  functions  must  certainly  accrue  at  some  time  or 
other;  for  there  can  be  no  doubt  that  we  come  in  time  to  be  thoroughly 
familiar  with  proportions  and  divisions  of  tonal  volume  by  mere  memory. 
As  we  have  seen,  all  our  sense  of  interval  is  founded  upon  that  ability. 
But  we  must  take  all  care  to  justify  our  musical  distinctions  by  powers 
that  dwell  in  the  stuff  of  tones  as  they  are  actually  presented  to  us, 


XXI]    RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY     195 

before  we  appeal  to  any  outlying  circumstances  or  to  habit.  We  must 
bring  our  engine  into  motion  with  its  own  steam,  ere  we  can  expect 
to  steady  its  action  by  its  acquired  momentum. 

The  results  of  the  above  suggestions  may  be  embodied  in  graphic 
form  (fig.  6).  The  height  of  the  curve  represents  in  some  fashion  the 
comparative  diaphony  or  paraphony  or  symphony  of  the  intervals 
below  it.  The  height  has  been  taken  in  most  case&  as  the  denominator 
of  the  fraction  by  which  the  volume  of  the  higher  tone  is  less  than  the 
lower,  thus  ^  or  thereby  (for  a  quarter  tone),  -^,  |,  ^,  i,  {,  ^,  similar 
values  hold  in  the  case  of  the  intervals  less  than  the  octave  from  just 
below  the  octave,  through  major  and  minor  seventh  to  major  sixth. 
The  intervals  on  either  side  of  the  fifth  must,  of  course,  have  high 
values,  sinking  in  the  tritone  perhaps  to  the  value  of  the  minor  seventh, 
and  in  the  minor  sixth  to  a  somewhat  lesser  quantity,  yet  still  greater 
than  that  of  the  major  sixth.  But  these  regions  cannot  yield  the  fierce 
'collisions'  that  we  get  near  the  octave  and  the  prime.  The  high  value 
of  the  augmented  fifth  lies  near  the  minor  sixth  as  the  quick  fall  of  the 
curve  in  some  way  indicates.  A  zero  value  may  be  assumed  to  lie  some- 
where in  the  paraphonic  zone^.  The  very  small  symphonic  zones  cannot 
properly  be  joined  to  the  rest  of  the  curve  by  lines;  for  there  is  no  gradual 
transition  from  them  to  their  neighbouring  diaphonies  through  an 
intermediate  paraphony.  So  it  would  be  useless  to  indicate  their  relative 
degree  of  symphony  by  vertical  position  in  the  figure.  There  is  a  transi- 
tion only  where  the  lines  are  drawn  out.  As  already  said,  the  ranking 
that  should  be  given  to  the  fourth  is  doubtful.  The  figure,  of  course, 
takes  no  note  of  any  differences  of  smoothness  that  may  be  due  to 
coincidence  of  partials  or  of  difference-tones,  as  these  are  only  subsidiary 
differences  that  would  be  ineffective  apart  from  the  differences  inherent 
in  the  tones  themselves  that  the  figure  attempts  to  indicate.  But  the 
beating  of  upper  partials  and  of  difference-tones  would  aggravate  the 
original  diaphony  of  the  primaries  most  of  all  in  the  immediate  neigh- 
bourhood of  the  great  consonances,  as  has  already  been  indicated^. 

•  According  to  Krueger  neutral  intervals  are  to  be  sought  "in  a  middle  region  of 
vibratory  ratios  close  beyond  the  sevens  intervale,"  i.e.  in  the  first  place  in  such  ratios 
a86:7,  4:9,  5:9,  7:9  (31,262). 

*  Our  figure  agrees  very  well  with  the  similar  figures  that  may  be  constructed  from 
the  experimental  ranking  of  the  intervals  of  the  octave  given  by  C.  F.  Malmbcrg  (3G,  m) 
for  the  'factors'  or  aspects  of  intervals  named  by  him  Smoothness  (relative  freedom 
from  beats).  Purity  (resultant  analogous  to  pure  tone)  and  Blending  (a  seeming  to  belong 
together,  to  agree.  The  curves  for  these  three  are  in  their  general  course  very  similar. 
Malmberg's  fourth  factor  of  Fusion  (a  tendency  to  merge  into  a  single  tone,  unanalysable) 

13—2 


196    RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY     [ch. 

Thereby  the  extent  of  the  zones  of  original  symphonies  has  perhaps 
been  somewhat  reduced;  and  the  point  of  optimal  symphony  has  been 
greatly  sharpened  and  emphasised  by  the  smooth  system  of  symphony 
throughout  difference-tones,  primaries,  and  partials  that  then  accrues. 
In  the  case  of  intervals  beyond  the  octave,  and,  still  more  so,  beyond 
the  double-octave,  the  ordinary  harmonic  partials  present  in  instru- 
mental tones  would  carry  forward  the  basis  of  original  paraphony 
which  we  have  established  for  the  first  octave  and  the  greater  part  of 
the  second.  And  the  beating  of  these  components  would  similarly 
mark  out  the  neighbourhood  of  these  distant  symphonies,  and  so  on. 
In  considering  the  need  to  appeal  at  some  time  to  the  memory  of 
volumic  proportions  and  their  systems  in  music  it  must  not  be  forgotten 
that  for  the  consonance  of  successive  tones  no  help  can  rightly  be 
sought  either  from  partials  or  from  difference-tones.  The  relations 
upon  which  we  have  founded  our  constructive  theory  are  the  only 
ones  that  then  remain  over.  Here  the  step  from  them  to  the  work  of 
memory  is  therefore  direct.  And  the  memory  of  volumic  proportions 
is,  as  we  know  from  vision  and  from  the  sense  of  time,  a  well  established 
and  well  trained  faculty. 

In  connexion  with  successive  tones  we  have  already  shown  how  the 
two  great  consonances  would  make  themselves  felt  even  in  music  that 
hardly  knew  such  a  thing  as  simultaneity  of  tones.  The  basis  of  these 
successive  consonances  is  exactly  the  same  as  that  of  the  simultaneous 
ones.  And  to  the  fifth  the  fourth  is  bound  almost  hand  and  foot,  as  a 
distinctive  grouping  of  tones.  For  the  other  intervals  less  than  the 
fifth — the  specially  emmelic  ones — no  clear  definition  is  given  in  mere 
succession.  Their  determination  will  therefore  be  decided  by  two  chief 
conditions  :  the  melodic  cogency  of  the  step,  which  is  the  greater  the 
smaller  the  step  (no  doubt  within  a  certain  limit) ;  and  a  basis  of  ease  of 
distinguishability  between  the  tones  of  the  step  which,  we  may  suppose, 
is  closely  akin  to  the  distinctiveness  of  pitches  which  produces  paraphony 

shows  itself  graphically  as  a  pure  distance-judgment,  correlated  to  separateness  of  pitches. 
Its  course  of  steady  decrease  from  minor  second  (maximum)  to  major  seventh  (minimum) 
is  broken  only  by  a  slight  rise  in  the  fifth  and  a  rise  to  greatest  maximum  in  the  octave. 
This  indicates  again  the  unitariness  of  these  intervals.  Probably  Blending  and  Purity 
are  expressions  for  the  same  thing  in  Malmberg's  tables.  He  also  found  that,  "when  by 
the  use  of  two  sets  of  tuning-forks,  the  just  intonation  was  compared  with  the  tempered 
intonation,  no  difference  of  ranking  of  the  intervals  large  enough  to  affect  the  order 
resulted  from  the  difference  in  temperament"  (p.  107). 

For  evidence  of  the  way  in  which  partials,  etc.,  mark  out  points  in  the  paraphonic 
zones  where  pure  tones  give  little  if  any  differences,  see  28,  491 1. 


XXI]    RETROSPECT  AND  THE  OUTLOOK  FOR  THEORY     197 

as  against  diaphony  in  simultaneous  tones.  It  is  in  agreement  with 
this  that  we  find  in  monophonic  or  homophonic  music  so  much  freedom 
and  variation  in  the  subdivision  of  the  two  '  tetrachords '  left  by  the 
fifth  and  fourth.  But  as  soon  as  polyphony  has  revealed  its  great 
preference  for  such  thirds  as  we  have  adopted  as  fundamental,  all  the 
melodic  or  successional  aspects  of  music  would  necessarily  come  under 
control.  And  we  have  already  shown  that  a  basis  exists  for  the 
(attitude  of)  apprehension  of  simultaneity  in  succession  that  we  find 
in  our  music  and  that  is  able  to  make  a  succession  of  tones  rather  a 
harmonic  whole  than  a  melody,  or  to  give  to  a  melodic  succession  the 
structural  form  and  the  atmosphere  of  a  harmony.  So  strong  in  fact 
is  this  colouring  of  harmony  in  many  of  our  melodies  that  various 
theorists  have  sought  for  a  purely  harmonic  foundation  for  melodic 
figuration, — a  procedure  which  is  surely  misleading.  This  harmonic 
tinge  in  melodies  only  appears  when  the  structure  of  the  sequence  is 
such  as  to  invite  that  sort  of  reminiscence  from  a  mind  already  full  of 
harmonic  experience.  But  it  never  appears  even  from  such  a  mind 
when  the  melody  proceeds  mainly  by  small  steps.  These  cogent  move- 
ments are  strong  enough  to  repel  any  incoherent  associations  towards 
dissonance  that  we  might  for  once  try  to  weave  into  them. 


CHAPTER  XXII 

SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION 

The  following  may  be  taken  as  a  summary  of  what  has  been  expounded 
in  the  previous  pages.  It  will  also  serve  as  a  sort  of  introduction  to  the 
positive  results  I  have  reached.  There  are  many  readers  who  will  be 
anxious  to  see  what  it  all  amounts  to  when  the  criticisms,  evidences 
and  authorities  have  been  laid  aside.  Besides,  no  time  should  be 
lost  in  bringing  to  the  succour  of  the  student  of  music  (and  perhaps 
of  harmony  in  particular)  what  may  be  firm  ground  for  his  intellectual 
efforts  with  that  art.  Until  now  he  has  been  fed  with  unexplained 
empiry  or  with  obviously  unreasonable  messes  of  harmonics,  both  of 
them  diets  of  no  sustenance  for  the  hungry  intellect.  The  teacher 
of  musical  theory  will  perhaps  be  able  to  follow  this  brief  sketch  of 
its  foundations  and  to  give  his  pupils  a  reasonable  and  continuous 
introduction  to  the  art.  For  that  reason  I  have  put  the  sketch  into  the 
form  of  an  outline  of  instructions,  in  which  the  topics  are  merely  arranged 
and  briefly  stated  in  their  systematic  order  without  being  expanded 
into  detail.  That  extension  the  teacher  will  readily  make  for  himself 
according  to  his  inclinations. 

1.  The  difference  between  tones  and  noises  is  fundamental.  Tones 
are  smooth,  regular,  balanced,  symmetrical;  noises  are  the  contrary. 

2.  Ordinary  musical  tones  are  really  blends  of  tones  (fundamental 
and  other  partials  or  harmonics).  Analysis  reduces  them,  by  methods 
that  are  already  generally  familiar,  to  a  single  series  of  fure  tones. 
Chap.  I. 

3.  Tones  differ  in  pitch,  whereby  they  fall  into  an  obvious  order. 
They  differ  also  in  volume  :  those  at  the  one  end  of  the  series  are  large, 
massive,  voluminous;  those  at  the  other  end  are  small  and  point  like. 
Between  the  two  ends  there  is  gradation  of  volume.  Tonal  volume 
we  feel  to  be  a  quantitative  difference  in  tones. 

4.  Volumes  commonly  consist  of  parts.  The  question  thus  presents 
itself  :  does  a  tone  consist  of  parts?  Have  different  tones  any  parts  in 
common? 

Consider  simultaneous  tones  first.  Play  c  with  d,  e,  f,  g,  a,  b,  c^, 
d^,  e^,  ...  {cd,  ce,  cf,  etc.).   Notice  : 


CH.  xxn]    SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION         199 

(a)  that  the  two  tones  never  seem  to  get  entirely  away  from  one 
another;  c  and  d  seem  to  confuse  each  other  very  much,  they  seem  to 
be  mixed  up  with  one  another  in  confusion ;  c  and  e  seem  to  be  less  '  in  a 
heap';  and  so  on  progressively;  c  and  d^  are  relatively  free  from  one 
another. 

(6)  that,  nevertheless,  the  series  of  pitches  of  d,  e,  /,  g,  etc.  is  the 
same  in  combination  with  c  as  it  is  when  these  tones  are  played  singly 
in  a  series;  the  series  stands  untouched  in  spite  of  the  fact  that  the 
tones  of  each  pair  never  become  quite  free  from  one  another. 

5.  Now  write  down  the  series  of  pitches  as  dots  (or  as  points 
separated  by  a  short  space,  representing  the  fact  that  between  c  and  d, 
for  example,  other  tones  could  be  distinguished — c#,  d\f,  quarter-tones 


■^b- 


-9-^ 


-A 


>-e- 


-d-*- 


.?.. 


Fig.  7.     Showing  the  relation  between  the  pitch  and  the  volume  of  the  tonal  series. 

In  this  diagram  the  volume  (length  of  line)  of  every  higher  tone  must  be  supposed  to  be 
projected  upon  the  volume  (line)  of  the  lower  tone.  They  have  been  separated  into 
an  ascending  series  only  for  clearness  of  exposition.  The  dots  represent  the  pitches 
of  tones. 

or  eighth-tones,  etc.).  Then  put  volumes  round  the  dots  to  satisfy 
paragraph  1  (tones  are  balanced,  symmetrical)  and  paragraph  ia  (no 
two  tone  volumes  are  entirely  clear  of  one  another).  It  will  be  found 
that  the  volume  of  every  tone  will  have  to  go  quite  up  to  the  right- 
hand  end  of  the  pitch  series  where  the  dots  for  the  highest  tones  lie. 
Thus  (cf.  Chap,  ii) :  Figure  7. 

6.  Repeat  this  series  cd,  ce,  cf,  etc.,  noting  this  time,  instead  of  the 
gradual  change  throughout  the  series,  the  features  that  distinguish 
cd  from  ce,  from  eg,  from  ce^,  etc.   For  this  purpose  it  may  be  necessary 


200  SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  [ch. 

to  compare  the  pairs  of  tones,  not  as  they  stand  in  a  series,  but  in  some 
other  order,  pair  against  pair.  In  cc^  and,  to  a  noticeably  less  extent, 
in  eg  we  observe  a  balance  or  symmetry  of  the  whole  sound  that  reminds 
us  of  the  appearance  of  a  single  tone. 

How  can  we  represent  this  with  the  lines  of  figure  7  ?  If  the  line  for 
the  higher  tone  of  the  pair  grows  progressively  smaller,  as  is  there 
indicated,  we  shall  at  a  certain  point  get  this  figure  (fig.  8)  where 
the  higher  tone's  volume  just  fills  the  upper  half  of  the  lower  tone's 
volume.  This  arrangement  would  surely  give  the  nearest  possible 
approximation  to  the  simplicity  and  balance  that  characterise  very 
smooth  (or  pure)  single  tones.  Hence  we  may  ascribe  it  to  the  octave. 
The  volumes  of  tones  an  octave  apart  would,  therefore,  be  to  one  another 
in  the  proportion  of  2  :  1. 

Another  form  of  balance  would  be  given  when  the  two  new  points 
of  the  upper  tone  lie  equally  far  on  either  side  of  the  middle  (pitch) 
point  of  the  lower  tone,  thus  balancing  each  other  around  it.  But  the 
whole  would  have  more  points  to  distinguish  it  from  the  single  (pure) 


Fig.  8.     Showing  the  relations  between  the  volumes  (and  pitch-points)  of  a  tone  and 

its  octave. 

tone  than  the  octave  has.  Hence  it  will  not  be  so  unitary  in  effect  as 
the  octave.  This  arrangement  (which  figure  7  shows  must  arrive  at 
a  certain  place  in  the  scheme)  we  may  ascribe  to  the  fifth.  And  the 
volumes  would  be  in  the  proportion  of  3  to  2  to  one  another. 

7.  Hence  we  now  know  precisely  how  simultaneous  tones  overlap, 
and  we  can  correct  (or  annotate,  as  the  case  may  be  according  to  the 
instructor's  procedure)  figure  7  to  that  effect.  The  volume  of  a  tone 
decreases  by  half  for  every  octave,  by  3  :  2  for  every  fifth.  (These 
proportions  have  actually  been  embodied  in  figure  7.)  These  proportions 
happen  to  be  the  same  as  those  that  hold  for  octave  and  fifth  between 
the  rates  of  the  physical  vibrations  required  for  these  intervals.  For  all 
the  other  intervals  the  ratios  of  their  volumes  follow  as  a  matter  of 
course  (just  as  do  the  ratios  of  the  rates  of  physical  vibration).  It  is 
the  balance  of  volumes  that  characterises  the  octave  and  the  fifth  as 
heard  intervals  in  respect  of  their  fusion;  they  form  a  sound  that  is 
more  like  one  tone  than  other  intervals  do  :  the  octave  most  and  the 
fifth  next.  The  other  intervals  do  not  differ  so  markedly  from  one 
another  in  their  appearance  of  balance  as  do  the  octave  and  fifth  from 


xxn]        SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  201 

one  another  and  from  all  the  rest.  But  they  have  been  graded  by  careful 
experiment  in  the  following  way  :  fourth,  major  3rd,  minor  3rd,  major 
6th,  minor  6th,  tritone,  major  2nd,  minor  7th,  minor  2nd,  and  major 
7th.  The  volumes  corresponding  to  these  intervals  seem  likewise  to 
grow  less  and  less  balanced.  Their  characteristic  points  come  more  and 
more  into  conflict  with  one  another.  But  for  a  fuller  account  of  these 
grades  we  must  await  certain  lines  of  information  to  be  given  below  (18). 
Chap.  m. 

8.  The  same  relations  can  be  established  between  successive  tones 
(in  the  matter  of  the  relative  positions  of  their  volumes)  as  hold  for 
simultaneous  tones.  Only  it  is  not  now  primarily  a  question  of  balance 
or  rivalry.  Sequences  of  tones  are  not  of  themselves  dissonances  (or 
consonances).  But  they  can  play  the  part  of  dissonances  (or  consonances, 
as  the  case  may  be),  in  music  (as  arpeggio  chords);  and  they  then  call 
for  the  same  treatment.  This  may  be  easily  explained  by  supposing 
that  upon  occasion  we  can  take  a  point  of  view  from  which  we  notice 
the  balance  of  parts  formed  by  two  successive  tones.  These  are 
apprehended  in  projection  against  one  another  as  parts  of  a  whole. 
In  fact  there  is  probably  a  special  charm  and  value  in  these  relations 
of  balance  or  unbalance,  because  they  are  not  accompanied  by  the 
actual  fusion  or  clashing  peculiar  to  simultaneous  tones.   Chap.  v. 

9.  Interval  is  not  the  same  thing  as  fusion  (or  as  consonance  and 
dissonance).  Thirds  and  sixths  are  similar  in  fusion,  but  very  different 
as  intervals.  Interval  is  the  constant  proportion  between  the  sizes  of 
the  volumes  of  any  two  tones  of  absolute  pitch.  The  same  proportions 
can  be  got  in  an  indefinite  number  of  lower  and  of  higher  tones.  Then 
the  two  volumes  are  proportionately  larger  or  smaller.   Chap.  vi. 

10.  The  musical  range  of  pitch  (approximately  Ag  to  &,  or  seven 
octaves  and  a  third)  is  the  range  of  difference  in  physical  vibratory 
rate  within  which,  in  the  volumes  of  the  tones  evoked,  there  is  a  constant 
volumic  proportion  (or  interval)  for  any  given  ratio  of  vibrations.  At 
the  upper  limit  of  the  musical  range  the  proportion  begins  to  be  too 
large  (e.g.  2  :  1  -f  instead  of  2  :  1  exactly  for  the  octave  ratio  of  vibra- 
tions), so  that  the  highest  tones  appear  to  be  somewhat  flat;  at  the 
lower  end  the  proportion  is  too  small  (1  :  2  -f-)  so  that  the  lowest  tones 
appear  not  to  be  low  enough,  or  as  low  as  they  should  be  according  to 
the  ratios  of  their  rates  of  physical  vibration.  (These  restrictions  seem 
clearly  to  be  due  to  the  limits  of  size  and  of  subdivision  of  the  responsive 
tissues  of  the  cochlea).   Chap,  vii, 

11.  The  predominance  of  the  central  point  of  the  volume  of  a  pure 


202  SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  [ch. 

tone  and  the  smooth  grading  of  intensities  round  that  pitch-point  set 
us  a  pattern  for  the  apprehension  of  tonal  masses  in  general.  Except 
in  so  far  as  melodic  interests  draw  us  away,  we  tend  to  apprehend  the 
mass  centrally,  i.e.  it  seems  to  have  the  pitch  of  its  lowest  component. 
When  the  melodic  interests  of  different  'parts'  or  voices  are  equally 
developed,  the  bass  will  be  the  most  effective.  For  this  reason,  also, 
we  reckon  intervals  always  upwards;  intervals  are  more  readily  learnt 
and  recognised  in  ascending  form;  descending  intervals  have  to  be 
learnt  to  a  greater  or  less  extent  as  something  fresh;  and  there  is  a 
similar  disconnexion  between  successive  and  simultaneous  intervals. 
Chap.  VIII. 

12.  The  harmonics  or  upper  partials  that  constitute  the  blend  of 
musical  tones  are  heard  synthetically :  i.e.  we  do  not  strive  to  concentrate 
the  attention  upon  them  separately  to  the  exclusion  of  the  rest  of  the 
blend ;  and  they  are  intentionally  kept  so  weak  that  this  will  be  difficult 
to  do.  We  hear  them  then  merely  as  a  pleasing  change  in  the  '  surface ' 
of  a  tone's  volume,  departures  from  the  ideal  smoothness  and  graded 
balance  of  pure  tone  that  add  interest  to  the  whole  while  leaving  its 
balance  and  smoothness  approximately  intact. 

Difference-tones  (of  which  the  chief  are  :  h-l,  and  2l-h)  occur  only 
when  two  primaries  are  sounded  together.  Their  presence  is  generally 
overlooked,  not  only  because  they  are  heard  synthetically  in  the  whole 
tone-mass,  giving  it  a  touch  of  lowness,  but  because  they  generally 
arise  only  within  the  ear  and  so  have  nothing  in  the  instrument  corre- 
sponding to  them,  and  because  they  appear  only  when  two  (primary) 
tones  are  sounded  together.  The  separation  of  these  primaries  analytic- 
ally is  a  task  that  exceeds  the  power  of  (perhaps)  most  people,  so  that 
it  is  little  wonder  that  the  difference-tones  pass  unobserved.  But  the 
chief  difference-tones  are  not  hard  to  hear. 

The  first  and  prevailing  attitude  adopted  by  a  listener  towards  the 
primary  tones  of  a  chord  is  likewise  that  of  synthesis,  which  is  founded 
upon  the  inevitable  overlapping  of  the  volumes  of  tones  (described 
above).  The  ease  of  analysis  that  is  possible  for  specially  gifted  ears 
cannot,  of  course,  annul  this  synthesis,  although  it  makes  the  ear  aware 
of  the  component  tones.  Practice  will  tend  towards  the  same  analytic 
ease,  which  naturally  directly  subserves  the  purposes  of  accurate  and 
beautiful  musical  production,  etc. 

Difficult  directions  of  synthesis  are  favoured  by  different  musical 
methods,  especially  by  melody  ('horizontal'  direction)  and  by  harmony 
('perpendicular'  direction).    Polyphonic  and  harmonic  music  differs  in 


XXII]         SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  203 

the  prevalence  in  each  of  one  of  these  methods  as  a  basis  of  structure  : 
melody  in  polyphony,  fusion  in  harmony.   Chap.  ix. 

13.  The  connexions  established  between  certain  chords,  whereby 
they  come  to  be  recognised  as  inversions  of  a  certain  chord,  are 
founded  upon  the  (volumic)  pattern  formed  by  the  volumes  of  the 
several  over-lapping  tones  (fig.  9).  This  pattern  is  not  obscured  or 
distorted  by  being  continued  upwards  or  downwards  into  the  neighbour- 
ing octaves.  And  the  continuous  pattern  thus  obtained  contains  all 
the  different  inversions  of  the  '  one '  chord. 

The  octave  is  the  only  interval  that  makes  possible  such  an  extension 
of  pattern  without  any  distortion  of  it.  The  fifth  is  quite  unsuitable 
for  this  purpose,  high  consonance  though  it  is.  This  peculiarity  of  the 
octave  is  referable  to  the  special  way  in  which  its  higher  tone  fits  into 
the  upper  half  of  its  lower  one. 

' 6' 

— 9' 

■ e' 


'• * • -t f.-t.  .^.•....*.. All 

V  X  Y       Z 

Pig.  9.     The  proportions  of  parts  in  the  'All'  line  are  repeated  from  VX  to  X  T  to  YZ  in 

reduced  form  (I  to  |  to  ^). 

But  the  connexion  thus  established  between  inversions  does  not  in 
any  way  annul  their  very  important  differences;  it  only  gives  them  a 
close  connexion  in  spite  of  these  differences.  And  this  connexion  is 
not  a  merely  formal  one,  based  upon  an  algebra  of  transposition;  it 
must  always  be  real,  i.e.  the  listener  must  feel  or  recognise  the  connexion 
of  the  tonal  patterns  with  one  another  or  with  this  pattern  that  includes 
them  all.     Chap.  x. 

[The  instructor  may  find  it  helpful  at  this  point  to  offer  the  following 
information  : 

The  organ  of  hearing  is  the  cochlea,  in  which  a  long  thin  membrane 
occurs,  called  the  basilar  membrane.  This  membrane  stands  in  connexion 


204  SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  [ch. 

with  a  parallel  strip  of  sensory  nerves.    The  cochlea  is  coiled  up  like 
a  snail's  shell  (without  the  turn  at  the  apex  of  it),  fig.  10. 

Every  note  that  reaches  the  ear  comes  first  upon  the  point  a  of  the 
membrane,  and  affects  a  length  of  it  that  is  propor- 
tional to  the  length  of  the  air- wave  for  that  note,  or 
inversely  proportional  to  the  number  of  air  vibrations 
per  second.  Thus  a  note  of  100  vibrations  will  afEect 
twice  as  long  a  strip  of  the  membrane  as  one  of 
200  vibrations,  but  each  begins  to  work  at  the 
point  a.  The  point  of  the  membrane  most  intensely 
affected  lies  at  the  middle  of  the  length  affected  by 
any  note,  and  from  that  point  to  the  ends  of  the 
part  affected  the  intensity  probably  decreases  evenly. 

Hence  flow  all  the  features  of  tones  enumerated 
in  the  preceding   paragraphs.     An   attempt  may  ^^  lo.^^^tline  of 
be  made  to  deduce  them  by  way  of  recapitula-      the  sensitive  strip  in 

J.-  a  human  cochlea. 

The  student  may  find  such  statements  as  thus  refer  to  a  physical 
membrane,  etc.,  not  only  acceptable  and  convincing,  but  also  preferable 
to  a  discussion  of  what  we  merely  hear : — in  spite  of  the  facts  (1)  that 
(with  proper  guidance)  tones  are  immediately  observable  by  him  in 
all  their  aspects,  (2)  that  he  has  possibly  never  even  heard  of  the  cochlea 
before,  and  probably  knows  nothing  of  it  by  direct  observation,  and 
(3)  that,  being  perhaps  a  music  student,  he  is  eager  to  enjoy  or  even  to 
create  works  whose  entire  essence  resides  in  sounds  aswe  merely  hear  them. 

People  of  this  age  are  credulous  with  regard  to  statements  of  a 
scientific  form  regarding  material  things,  while  they  will  hardly  even 
believe  that  what  they  can  plainly  hear  or  see  or  in  general  have  before 
or  in  their  minds  can  be  properly  observed  or  made  the  subject  of 
scientific  study  and  proof.  When  invited  to  note  a  description  of  mere 
experiences  (e.g.  sounds  and  tones)  and  to  follow  a  train  of  reasoning 
about  them,  they  feel  that  they  are  being  deluded  or  beguiled  into 
mystical  fancies,  if  the  conclusions  attained  embody  results  that  surprise 
them.  But  there  is  no  reason  why  we  should  be  unable  to  observe 
aspects  of  experiences  (e.g.  of  tones)  and  by  reasoning  to  infer  things 
about  them  that  we  had  not  previously  observed  or  known. 

Even  w^riters  on  the  science  of  music  seem  not  uncommonly  to  be 
almost  afraid  to  trust  the  verdict  of  the  ear,  as  if  the  system  of  these 
verdicts  that  accumulate  in  the  course  of  time  would  likely  «how  itself 
to  be  erratic  and  confused,  uncontrolled  by  any  sort  of  law  and  order.] 


xxn]        SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  205 

14.  Melody  is  constituted  not  by  a  mere  succession  of  tones,  but 
by  a  motional  phenomenon  or  connexion  between  tones  which  supervenes 
upon  their  sequence,  provided  their  pitch  difference  and  their  distance 
in  time  is  not  too  great.  This  motional  connexion  is  somewhat  the 
same  as  the  connexion  that  turns  the  successive  stationary  pictures  of 
the  cinema  into  a  continuous  picture  in  which  the  parts  move.  Many 
useful  analogies  can  be  drawn  between  these  melodic  and  cinematic 
motions.  But  they  must  not  be  exaggerated.  Melody  is  not  a  slurring 
or  gliding  tone;  it  is  a  much  finer  kind  of  motional  connexion  than 
that,  and  it  supervenes  even  when  the  sequent  tones  do  not  glide  in 
pitch  at  all. 

The  motional  connexion  between  tones  is  the  more  obvious  the 
nearer  the  tones  lie  to  one  another.  The  consonance  of  the  interval 
included  in  a  leap  probably  improves  the  connexion.  So  does  any  other 
influence  that  strongly  suggests  a  next  tone,  especially  a  near  one. 
Hence  arises,  for  example,  the  powerful  tendency  of  the  'leading' 
tone  to  the  tonic. 

15.  As  soon  as  music  passes  the  homophonic  stage  of  a  single 
melody  with  a  rudimentary  accompaniment,  the  chief  problem  is  how 
to  make  two  or  more  melodies  move  clearly  alongside  one  another 
without  their  arresting  or  confusing  one  another's  motions.  This  is 
the  problem  of  polyphony,  and  music  in  two  or  more  voices  is  well 
termed  polyphonic.  It  might  also  be  called  polymelodic.  A  clearer 
term  is  afforded  by  the  word  'paraphony,'  which  associates  the  notion 
with  the  familiar  word  'polyphony,'  but  indicates  specially  that  the 
voices  have  to  move  alongside  (para)  one  another,  without  any  mutual 
disturbance  or  confusion  (p.  155  ff.). 

Harmonic  does  not  differ  from  polyphonic  music,  as  the  latter  terra 
might  suggest,  in  not  satisfying  the  needs  of  paraphony  and  so  in  not 
being  a  combination  of  voices,  but  in  being  rather  a  new  way  of  putting 
tones  together.  In  so  far  as  paraphony  is  concerned  there  is  practically 
no  difference  between  the  two  styles.  The  difference  lies  in  the  aspect 
of  tone  combinations  that  is  made  the  ground  of  the  artistic  synthesis. 
In  polyphony  the  melodic  outlines  or  figures  of  each  voice  are  the 
elements  of  artistic  construction:  each  melody  must  be  either  at  least 
tense,  cogent,  and  expressive,  or  at  most  skilfully  thematised,  commonly 
in  close  relation  to  the  accompanying  melodies.  In  harmonic  music 
this  interest  has  been  largely  neglected  in  favour  of  a  great  concern 
for  the  tonal  patterns  created  at  their  moments  of  simultaneity.  These 
patterns  (or  textures  or  colours — all  these  terms  suggest  a  modification 


206  SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  [ch. 

of  the  'surface'  of  tonal  masses),  are  now  made  the  chief  ground  or 
object  of  artistic  workmanship.  Only  one  voice — commonly  the  highest 
— is  thematised,  the  others  hardly  at  all.  At  times  none  may  be  so; 
then  the  music  wafts  along  in  breezes  of  harmonic  surface  or  colour. 
But  the  needs  of  paraphony  must  always  be  satisfied.  And  so  we  must 
recognise  that  all  music  is  primarily  and  essentially  paraphonic  and 
only  in  the  second  place  either  polyphonic  (in  the  historical  sense)  or 
harmonic  (or  ultra-harmonic).    Chap.  xix. 

16.  As  far  as  circumstances  permit,  simultaneous  melodies  form 
themselves,  or  present  themselves  formed  to  our  attention,  just  as  do 
single  melodies.  We  do  not  need  to  attend  to  them  and  thereby  to 
combine  their  tones  into  a  sequence  by  an  inner  act  of  mind.  In  fact 
we  do  not  know  which  tones  belong  to  the  one  or  other  melody  until 
we  find  them  incorporated  in  it.  The  melodies  flow  beside  one  another 
like  the  streams  of  raindrops  that  run  down  a  window  pane  in  devious 
neighbourly  courses.  The  essential  and  inevitable  overlapping  of  tonal 
volumes  does  not  seem  to  create  any  general  disturbance  of  paraphony. 
Hence  we  may  infer  that  melodic  motion  holds  rather  between  the 
pitch  centres  of  these  volumes  than  between  the  volumes  as  a  whole. 

This  primary  freedom  of  paraphony  is  characteristic  of  the  intervals 
of  thirds  and  sixths.   Chap.  xiii. 

17.  The  other  intervals  offer  less  freedom  to  paraphonic  flow.  In 
relation  to  the  series  of  fusions  (given  in  paragraph  7  above)  we  find 
towards  the  consonant  end  an  increasing  effect  of  (symphonic)  obscurity 
and  arrest;  towards  the  dissonant  end  an  increase  of  (diaphonic)  con- 
fusion and  restlessness.  '  Symphony '  expresses  the  apparent  unitariness 
of  the  interval's  mass  of  sound,  its  approximation  to  the  balance  and 
unity  of  a  single  tone.  Compare  the  Greek  definition  of  'symphony.' 
'Diaphony'  indicates  the  apparent  inter-penetration  of  the  tones,  by 
which  they  obscure  one  another's  pitches  and  outlines,  and  so  suggest 
a  movement  to  other  neighbouring  tones  that  will  give  either  greater 
balance  or  easier  paraphony.  Symphony  (and  unison,  i.e.  a  single  tone 
as  the  momentary  identity  of  two  voices)  therefore  creates  a  break 
of  melodic  flow  and  a  feeling  of  arrest,  diaphony  tends  to  obscure  the 
melodic  lines  and  suggests  transition  to  more  peaceful  terms.  Chap,  xviii. 

18.  We  have  already  seen  how  the  symphonies  of  the  octave  and 
fifth  are  created  by  the  peculiar  way  the  volumes  of  these  tones  fit 
into  one  another  so  as  to  appear  like  one  balanced  whole  or  like  one 
tone.  In  diaphony  there  is  a  collision  or  rivalry  of  the  defining  parts 
of  the  volumes  of  tones  due  to  their  proximity  to  one  another  (thus 


XXII]        SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  207 

all  the  diaphonies  neighbouring  on  the  octave  and  on  the  prime  or 
unison)  or  to  their  failure  by  a  little  to  yield  the  balance  of  the  fifth 
(thus  the  small  zone  of  diaphony  on  either  side  of  the  fifth).  In  paraphony 
the  defining  points  of  the  two  volumes  stand  well  clear  of  one  another 
so  that  the  two  lines  are  easily  apprehended  as  distinct  even  though 
simultaneous.  These  relations  can  be  traced  more  or  less  distinctly 
throughout  the  greater  part  of  the  second  octave,  especially  up  to  the 
twelfth.  The  irregular  position  of  the  fourth  in  this  scheme  corresponds 
with  the  somewhat  imstable  character  of  that  interval  in  music. 
Chap.  XXI. 

19.  The  effects  of  symphony  and  diaphony  are,  of  course,  worse 
when  two  intervals  of  the  same  species  follow  one  another  than  in  a 
single  interval  of  that  kind.  Hence  sequences  or  '  consecutives '  are 
forbidden, — the  more  stringently  the  more  symphonic  or  diaphonic 
the  interval  concerned  originally  is  :  octaves  (or  unisons)  most,  fifths 
less,  fourths  only  a  little;  minor  seconds  and  major  sevenths  more 
than  major  seconds  and  minor  sevenths;  or  minor  seventh  and  tritone 
perhaps  less  than  other  dissonances.   Chap.  xii. 

But  single  intervals  are  forbidden  too,  or  are  at  least  inadvisable, 
unless  they  are  approached  in  a  way  that  counteracts  or  obscures  their 
own  symphonic  or  diaphonic  effect.  Chap.  xv. 

20.  Of  these  counteracting  or  obscuring  methods  the  most  important 
is  the  use  of  contrary  motion  in  approaching  symphonies  and  diaphonies. 
This  has  the  effect  of  annulling  the  unitariness  and  arrest  peculiar  to 
symphony  and  the  obscurity  and  confusion  of  diaphony.  The  lines  of 
motional  connexion,  proceeding  in  contrary  directions  either  inwards 
or  outwards  from  the  pitches  of  one  interval  to  those  of  the  next,  stand 
clear  of  one  another  and  are  therefore  paraphonic. 

Oblique  motion  has  a  kindred  function.  The  stationary  voice 
involves  only  a  renewal  of  the  sound  already  heard.  The  other  voice 
has  then  almost  the  same  freedom  of  action  as  a  single  melody.  There 
is  no  arrest  or  confusion  between  the  two  voices — the  stationary  and 
the  moving  one, — because  the  one  voice  moves  so  clearly  and  distinctly. 

Similar  motion  probably  leaves  the  intervals  quite  unaffected,  so 
that  their  own  naturally  inherent  characteristics  of  unity  and  balance 
(symphony)  or  confusion  and  rivalry  (diaphony)  emerge  unmodified. 
The  reason  why  similar  motion  is  so  often  forbidden  in  musical  con- 
struction is  not  that  such  motion  is  a  bad  method  in  itself,  but  because 
it  is  ineffective  to  remove  the  undesirable  characteristics  of  the  intervals 
concerned.    It  is  the  interval  that  is  bad,   not  the  similar  motion. 


208  SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  [ch. 

Contrary  motion  is  specially  good  because  it  annuls  the  confusing  effects 
of  symphony  and  diaphony. 

Melodies  should  not  cross  or  overlap  because  the  motional  connexion 
that  constitutes  melody  tends  to  establish  itself  through  the  nearest  or 
easiest  steps.  Thus  the  individuality  of  melodies  tends  to  be  broken, 
unless,  of  course,  a  special  character  is  given  to  each  by  its  tone  blend 
(or  timbre)  or  by  other  means  such  as  the  special  figures  or  themes 
contained  in  the  melodies,  and  so  on.  Chap.  xx. 

21.  Of  the  tones  of  an  interval  the  lower  is  on  the  whole  the  pre- 
dominant one.  If  other  tones  are  inserted  between  the  first  two,  they 
are  less  prominent  than  either  of  the  others.  Thus  we  can  readily 
establish  a  grading  of  prominence  or  exposure  amongst  the  pairs  of 
voices  of  a  polyphonic  work.  This  grading  is  :  B-S,  B-A  and  B-T,  S-A 
and  S-T,  A-T,  where  the  letters  stand  for  Bass,  Tenor,  etc.  (p.  102  f.) 
Whatever  appears  in  any  pair  of  voices  is  the  more  effective  in  proportion 
to  the  exposure  given  by  the  voices.  If  the  effect  is  bad,  it  will  be  worse 
than  it  otherwise  would  be ;  if  it  is  good,  it  will  be  better.  Hence  arises 
the  gradation  of  prohibitions  according  to  the  pair  of  voices  to  which 
they  refer.     Chaps,  xii,  xv. 

The  prominence  of  the  inner  parts  will  naturally  be  less  the  greater 
their  total  number.  Hence  we  find  a  gradual  relaxation  of  prohibitions 
with  the  increase  in  the  number  of  parts.  Even  the  outer  parts  are 
less  prominent  when  there  are  many  altogether  than  when  there  are 
few,  so  that  the  relaxation  of  prohibitions  is  quite  general  in  its  degree. 
Each  melodic  rivulet  is  less  clear  and  distinct  when  there  are  many  of 
them.  The  common  preference  for  four-part  music  probably  indicates 
that  in  four  parts  a  fair  balance  is  attained  between  (a)  the  disadvantage 
of  the  unvaried  exposure  of  all  details  and  the  thinness  of  interest  in 
two  or  three  voices,  and  (6)  the  disadvantage  of  the  loss  of  distinction 
of  details  and  the  overfeeding  of  the  interest  by  more  voices  than  four. 
Chap.  XX. 

22.  Contrary  motion  is  not  the  only  means  of  reducing  symphony 
and  diaphony  to  paraphony.  Any  means  is  effective  in  so  far  as  it 
strengthens  the  melodic  connexions  leading  through  the  unfavourable 
interval. 

An  obvious  device  consists  in  making  the  motional  tension  of  each 
part  specially  strong.  That  tension,  as  we  saw,  is  the  greater  the  nearer 
the  tones  lie  to  one  another  in  pitch.  Hence  at  critical  moments  when 
an  unparaphonic  interval  has  to  be  passed  without  the  help  of  contrary 
or  oblique  motion,  we  may  help — it  is  the  artist's  problem  whether 


xxn]        SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  209 

that  help  is  at  any  time  sufl&cient — the  motion  by  step  rather  than  by 
leap  in  one  voice  or  in  both.  As  the  highest  voice  is  in  harmonic  music 
usually  the  most  distinctively  melodic  (or  thematic),  the  step  will  be 
more  effective  in  it.  Here  again  we  can  easily  set  up  a  gradation  of  cases  : 
worst — leaps  in  both  voices,  step  in  lowest  and  leap  in  highest,  step 
in  highest  and  leap  in  lowest,  steps  in  both  voices — best.  If  one  of  the 
voices  be  an  inner  voice,  the  last  degree  of  this  series  would  probably 
be  an  unnecessary  embellishment  (p.  128  f.). 

23.  The  two  preceding  paragraphs  (21,  22)  indicate  that  neither 
paraphony  nor  polyphony  is  to  be  supposed  to  be  limited  to  two  voices 
or  to  any  definite  number  of  them.  The  problems  of  paraphony  run 
continuously  into  any  number  of  parts.  The  effects  are  complicated 
only  by  the  greater  number  of  intervals  that  then  appear  and  the 
difficulty  of  regulating  them  all  together  to  the  desired  effect, — either 
melodic  motion  for  a  specified  length  of  time  or  arrest  of  it  at  a  certain 
time  or  a  variable  degree  of  this  arrestive  effect.  The  difficulty  of 
regulating  many  parts  at  once  is  due  to  the  different  symphonies  and 
diaphonies  that  inevitably  appear  when  a  column  of  paraphonic  intervals 
is  formed,  e.g.  ceg  or  ce^g,  etc.  (We  suppose  here  logically  that  polyphony 
sets  out  from  an  attempt  to  combine  several  paraphonic  intervals 
together.  Really  the  general  course  of  progress  went  probably  rather 
from  the  greater  consonances,  which  first  drew  the  greater  attention 
to  themaelves  and  were  used  to  embellish  a  single  melody  during  its 
course  and  without  themselves  forming  another  melody.  The  slowness 
of  early  progress  was  probably  largely  due  to  the  obstruction  natural 
to  this  starting  point.  The  consonances  are  the  easiest  intervals  to  find 
but  the  more  difficult  to  use  artistically.) 

The  resultant  effect  of  such  combination  depends  upon  factors 
already  enumerated,  especially  the  nature  of  the  interval  formed  and 
the  exposure  given  to  it  by  the  voices  it  stands  in.  Thus  we  can  arrange 
the  various  chords  that  involve  the  same  set  of  pitch-names  according 
to  the  greater  stability  of  effect.  For  example,  of  chords  involving 
the  tones  c,  e,  and  g,  ceg  is  the  stablest  and  most  arrestive,  because 
it  contains  a  fifth  (consonance  of  second  grade)  in  the  outer  voices 
and  the  most  fused  paraphony  (major  third)  in  the  outer  voices.  Of 
the  two  inversions  the  second  gc^e^  is  better  than  the  first  egc^,  because 
the  former  has  major  paraphonies  for  the  latter's  minor  ones  and  the 
fourth  is  less  noticeable  in  the  upper  voices.  Similarly  in  chords  con- 
taining the  tones  6,  d  and  /  the  worst  or  most  confusing  arrangement 
is  hd}J^,  then  fbd^,  while  dfb  is  the  best. 

W.  W.  M.  1* 


210  SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  [ch. 

24,  Because  of  the  general  trend  of  these  differences  and  also  because 
of  the  theoretical  interest  of  columns  of  thirds  (major  or  minor),  the 
arrangement  of  the  tones  of  a  chord  that  gives  a  column  of  thirds 
has  been  called  its  root-position.  We  can  readily  think  into  that  terra 
the  notion  of  greater  stability,  arrestiveness,  or  paraphony.  But  it 
is  wrong  to  suppose  that  any  chord  has  been  in  any  sense  generated 
from  any  one  tone,  as  the  older  theories  that  operated  exclusively  with 
harmonics  supposed.  But  a  small  chord,  e.g.  hd^f^  may  well  come  to 
be  felt  to  be  a  part  of  another  larger  chord  {gbd]f^)  in  so  far  as  the  latter 
(or  its  'pattern')  includes  the  former  and  comes  to  be  so  important 
for  music  generally  as  to  attach  the  former  to  itself  as  a  mere  part 
or  reminiscence.  But  this  connexion,  if  it  is  to  be  valid,  must  be  really 
felt. 

When  symphonic  intervals  are  combined  with  diaphonic,  we  find 
that  the  loss  of  distinction  due  to  the  great  unitariness  of  the  former 
is  largely  annulled  by  the  loss  of  distinction  due  to  the  confusion  intro- 
duced by  the  latter.  Consequently  single  and  second  symphonies  (octaves 
or  fifths)  may  pass  unguarded  (without  the  help  of  contrary  motion,  etc.) 
in  discords. 

25.  A  special  paraphonic  difficulty  is  created  by  the  interval  of 
the  fourth,  especially  when  it  is  exposed  by  resting  upon  the  centre 
of  a  sound  mass — the  bass.  The  fourth  is  not  a  dissonance,  as  has 
often  been  asserted.  It  is  what  we  already  know  it  to  be — the  lowest 
grade  of  symphony.  But  especially  when  it  is  exposed  rhythmically  as 
well  as  phonically,  it  seems  to  call  for  or  to  suggest  the  neighbouring 
major  third,  and  so  to  claim  a  resolution,  as  do  the  dissonances.  If  this 
suggestion  is  to  be  avoided,  exposure  of  the  interval  must  be  reduced 
as  far  as  possible,  and  the  melodic  line  leading  through  the  chord  in 
which  the  fourth  appears  must  be  made  as  cogent  as  possible,  especially 
the  line  through  the  bass.  If  the  suggestion  towards  the  third  is  welcome, 
it  is  increased  by  rhythmical  exposure  and  so  a  special  progression — a 
(symphonic)  cadence  (or  general  arrest  of  melodic  flow) — is  attained. 
Chaps,  XVI,  XX. 

The  case  of  the  bass-fourth  is  probably  only  one  of  a  class  of  similar 
cases.  Another  example  seems  to  be  given  by  the  minor  sixth,  which 
has  been  held  to  be  dissonance,  although  it  is  now  generally  treated 
as  a  consonance.  But  the  interval  identical  with  it  in  equal  temperament 
— the  augmented  fifth — is  always  a  dissonance,  probably  because  the 
augmented  triad  so  strongly  suggests  the  major  triad  that  the  former 
appears  as  a  distortion  of  the  latter. 


xxn]       SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  211 

This  reason  for  the  difficulty  of  the  fourth  from  the  bass  explains 
why  the  effect  of  such  a  fourth  is  not  obliterated  in  a  discord,  while  a 
second  fourth  in  the  same  voices  (consecutive  fourths)  is  relieved,  if 
either  the  second  (or  even  the  first)  stands  in  a  discord.  In  the  latter 
case  the  fourth  acts  merely  as  the  symphony  it  is ;  in  the  former  its  effect 
is  due  to  the  call  for  the  major  third  excited  by  its  exposure  on  the  bass. 

26.  The  tendencies  and  possibilities  of  resolution  of  dissonances  are 
not  so  much  the  result  of  any  sort  of  connexion  between  different 
intervals,  as  far  more  probably  a  general  expression  of  the  most  cogent 
melodic  movements  that  will  satisfy  the  propulsion  of  melodic  move- 
ment that  is  characteristic  of  diaphony.  The  most  cogent  movement  is 
in  the  first  place  the  shortest  (in  difference  of  pitch),  but  due  regard 
must  be  paid  to  the  limitations  imposed  by  key  structure,  the  paraphonic 
nature  of  the  resulting  (following)  interval  or  of  the  whole  chord  in 
which  it  stands,  thematic  form,  change  of  key,  etc.  Apart  from  these 
determining  conditions  there  are  probably  no  limits  to  the  possibilities 
of  resolution. 

27.  It  is  not  to  be  supposed  that  chords  are  essentially  different 
from  intervals.  Both  of  them  are — in  music — primarily  paraphonic 
structures,  which  differ  only  in  the  number  of  voices  involved.  The 
putting  together  of  three  voices  is  only  an  extension  of  the  task  of 
combining  two  melodies,  not  a  new  kind  of  task.  But  we  learn  to  look 
upon  the  structures  formed  by  three  or  more  voices  as  typical  volumic 
patterns,  which  we  reduce  to  a  few  comprehensive  types.  The  relation 
of  inversion,  for  example,  forms  a  very  important  step  in  this  reduction. 
So  accustomed  do  we  become  to  this  procedure  that  we  largely  cease 
to  be  able  to  look  upon  intervals  (of  two  tones)  except  as  fragments 
of  chords.  Nevertheless  there  was  a  time  (for  music  generally  and  for 
each  musical  person)  when  intervals  had  not  yet  come  into  these 
relations.  The  final  result  of  long  experience  with  the  most  frequent 
types  of  chords  is  the  formation  of  a  system  of  chords  and  their  relations 
and  progressions,  commonly  known  as  Harmony. 

The  artistic  concentration  of  attention  upon  harmonic  rather  than 
upon  melodic  form  and  relations  leads  to  a  special  systematisation  of 
chords  by  means  of  which  they  are  grouped  round  a  centre  (tonality) 
with  parallel  sub-centres  (keys).  Tonality  reacts  upon  the  scale,  moulding 
it  so  as  to  make  it  yield  the  biggest  and  most  coherent  system  of  chorda 
possible.  From  this  diatonic  harmony  the  system  is  extended  gradually 
to  a  system  of  chromatic  harmony  and  to  other  types  that  are  at  present 
insufficiently  defined. 

14—2 


212  SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  [ch. 

28.  The  development  of  tonality  raises  into  importance  the  chords 
that  are  related  to  one  another  in  certain  special  ways,  especially 
those  that  stand  at  the  interval  of  a  fifth  or  fourth  from  one  another. 
Here  we  have  in  a  new  form  the  relation  of  balance  characteristic  of 
the  symphonies;  the  octave  virtually  drops  out  because  the  repetition 
of  a  chord  an  octave  away  is — in  all  but  its  'brightness'  or  volumic 
size — the  same  chord:  or  the  octave  is  just  the  greatest  balance,  if  you 
like.  The  fifth  (or  fourth)  is  the  next,  and  therefore  the  main,  pivot 
of  all  tonality.  These  points  of  tonality  are  also  the  only  ones  that 
repeat  a  chord  exactly  without  any  distortion.  (Possibly  this  balance 
of  chordal  relations  at  the  fifths  is  responsible  for  the  final  form  of  the 
scale.)  So  we  come  to  think  these  points  easily  together.  And  their 
connexions  can  thus  combine  with  the  circumstances  already  indicated 
to  counteract  the  effects  of  symphonic  and  diaphonic  intervals  towards 
any  special  end. 

There  is  some  evidence  that  this  power  of  harmonic  connexion 
varies  with  its  closeness,  being  greatest  for  the  fifth-fourth,  next  for 
the  third-sixth,  and  least  for  the  second-seventh  (or  no)  relation.  But 
the  matter  probably  requires  further  justification  before  it  can  be  taken 
in  this  general  form.  For  the  first  grade  of  harmonic  connexion  there 
is  no  doubt  of  the  existence  of  a  modifying  power. 

29.  It  must  not  be  thought  that  the  laws  of  primitive  music  are 
annulled  by  its  development.  They  are  rather  merely  complicated  and 
fulfilled.  Rules  of  harmony  that  bind  one  age  are  not  disregarded  in 
the  next.  Nor  do  we  merely  learn  to  tolerate  what  a  previous  age 
forbade.  We  rather  learn  to  annul  or  to  transmute  the  effects  that 
once  created  strain,  by  bringing  to  bear  upon  them  influences  that  the 
previous  age  had  not  discovered,  but  that  are  essentially  the  natural 
products  of  musical  development,  or,  better,  of  tonal  artistry.  We 
bring  the  once  jarring  phase  into  a  setting  in  which  it  forms  a  proper 
extension  of  those  very  laws  that  were  previously  the  just  ground  and 
reason  of  its  prohibition.  Music  is  an  orderly  realm  of  beauty,  intelligible 
in  every  aspect;  it  is  not  a  temporary  code  of  arbitrary  preferences 
that  extol  themselves  as  art. 

The  main  thesis  propounded  in  this  book,  so  far  as  the  actual 
structure  of  music  itself  is  concerned,  is  embodied  in  the  paragraphs 
of  the  preceding  summary  from  no.  14  onwards.  It  may  be  briefly 
stated  here  : 

The  fount  and  origin  of  probably  all  music  whatever  is  melodic 


xxn]        SYNOPSIS  OR  OUTLINES  OF  INSTRUCTION  213 

movement,  or  simply,  melody  (not  limited,  however,  to  thematised 
melody).  Primitive  music  is  monomelodic,  embellished  by  reduplication 
at  some  interval  or  with  irregular  heterophonic  accompaniment.  Modern 
music,  whether  it  be  classed  as  polyphonic  or  harmonic,  belongs  to  a 
great  era  of  polyphony,  of  which  the  essential  problem  is  the  construction 
of  concurrent  melodic  streams  that  will  leave  each  other's  motions 
unimpaired  and  produce  effects  of  arrest  as  they  may  be  desired.  The 
solution  of  this  problem  embraces  the  solution  of  the  nature  of  similar, 
oblique,  and  contrary  motion,  of  the  prohibition  of  consecutive  intervals 
of  the  same  kind  as  well  as  of  'hidden'  consecutives,  of  the  puzzle  of 
the  fourth  from  the  bass,  of  chordal  structure  in  general  and  of  the 
differences  from  one  another  of  a  chord  and  its  inversions,  of  the  need 
for  resolution  of  discords  and  its  tendencies  and  possibilities,  etc. 

Or :  it  is  commonly  held  that  our  music  is  "the  concord  of  sweet 
sounds,"  a  structure  of  which  the  fundamental  material  and  ornament 
is  consonance  in  its  special  or  more  perfect  grades.  We  might  rather 
say  that  music  is  a  'concourse'  of  sweet  sounds,  in  the  literal  sense  of 
the  word,  a  structure  in  which  movement  or  melody  is  the  fundamental 
material.  There  is  rather  an  opposition  than  a  kinship  between  the 
functions  of  melody  and  of  consonance.  The  latter  is  the  principle 
of  arrest,  inimical  to  the  free  course  of  melody.  Music  has  been  created 
rather  in  spite  of  consonance  than  by  its  help. 

Thus  the  Greek  definition  of  consonance  is  upheld  both  in  its  connota- 
tion and  denotation.  Symphony  reveals  a  loss  of  distinction  of  tones 
in  apparent  unity.  The  neutral  intervals  of  thirds  and  sixths  must  be 
recognised  as  paraphonic;  their  tones  are  freely  distinguishable.  In 
dissonance  or  diaphony  we  find  a  loss  of  distinction  in  confusion  or 
harsh  collision.  Thus  we  give  new  significance  and  confirmation  to 
the  classification  of  intervals  proposed  by  Gaudentius.  Music  progresses 
by  greater  differentiation  between  intervals  and  by  the  discovery  of 
further  factors  that  modify  their  paraphonic  properties,  not  by  a 
gradual  lowering  of  the  border  line  of  consonance. 


CHAPTER  XXIII 

THE  OBJECTIVITY  OF  BEAUTY 

Eemares  have  been  made  at  various  points  throughout  the  preceding 
chapters  in  deprecation  of  the  common  tendency  to  look  upon  beauty 
as  a  matter  of  convention,  or  of  merely  subjective  personal  fancy, 
such  as  prevails  in  one  person  or  age  and  may  be  rejected  in  another. 
One  feels  a  certain  reluctance  towards  passing  from  the  study  of  the 
foundations  of  musical  beauty  in  detail  to  the  exposition  or  defence 
of  a  general  thesis  such  as  that  of  the  objectivity  of  beauty.  Those  who 
study  the  detail  closely  will  surely  come  of  themselves  upon  the  general 
point  of  view,  and  will  feel  its  cogency.  And  there  is  a  great  pleasure 
in  thinking  out  such  conclusions  and  in  seeing  how  they  extend  their 
satisfying  graces  into  the  whole  of  one's  world.  On  the  other  hand 
there  is  a  certain  amount  of  laborious  thought  involved  in  these  visions, 
the  communication  of  which  may  help  to  lighten  the  work  of  others. 
In  any  case,  every  one  of  us  has  his  own  particular  world  to  fill  with 
strength  and  joy,  so  that  there  is  no  real  risk  of  robbing  any  man  of 
the  pleasant  excitement  of  that  task. 

The  age  we  live  in  is  in  many  ways  strongly  coloured  by  the  outlook 
of  subjectivism.  We  have  passed  through  the  time  in  which  the  first 
great  generalisations  of  biology  were  made.  Each  type  of  creature  has 
been  shown  to  be  the  product  of  innumerable  influences  that  not  only 
work  upon  it  now,  encouraging  one  individual  and  eliminating  the 
other,  but  that  played  in  the  same  way  upon  the  infinite  series  of  its 
ancestors.  The  choosings  of  chance  have  made  us  as  similar  or  as 
different  as  we  actually  are.  Our  members,  their  forms  and  functions, 
our  innermost  structure  and  processes,  in  short  every  infinitesimal  part 
of  us  is  to  a  greater  or  less  degree  peculiar  to  us,  unique,  sifted  ofE  from 
the  great  drift  of  life's  variations,  and  stamped  'individual.'  How, 
then,  can  we  expect  to  be  so  fundamentally  akin  that  the  world  should 
be  the  same  even  for  two  of  us?  Why,  our  very  eyes,  our  brain,  and 
the  minute  chemistry  that  is  its  change,  have  been  selected  from  myriads 
of  possibihties !  Even  beauty  has  been  held  to  be  merely  contributory 
to   life's   functions,   a  subjective    accident   enlisted  in    the    work    of 


CH.  xxiii]  THE  OBJECTIVITY  OF  BEAUTY  215 

reproduction  to  be  a  symbol  of  the  fitness  of  male  and  of  female  to  one 
another.  And  we  make  our  arts  out  of  these  symbols  by  wandering 
from  the  straight  path  of  life's  purposes  and  by  perverting  a  good 
function  to  a  useless  play.  So  we  come  to  look  upon  art  as  perversion 
and  upon  the  artist  as  an  unpractical  trifler,  stifling  the  true  functions 
of  his  mind — which  should  serve  the  great  ends  of  life — by  combining 
them  into  vain  display. 

But  the  artist  reaps  the  inevitable  reward  of  his  perversion.  He 
flatters  himself  with  the  thought  of  progress  :  in  the  end  so  many 
revolutions  are  wrought  in  his  art  that  the  nothingness  and  unreaUty 
of  all  of  them  become  too  obvious  to  ignore.  Art  then  comes  to  be 
the  whim  of  an  age,  the  latest  device  to  thrill  and  to  dazzle  a  mind 
that  is  'real'  only  when  it  passively  acts  as  a  drop  in  the  incoherent 
drift  of  life,  but  that,  blowing  itself  out  to  some  semblance  of  intrinsic 
worth  and  orderliness,  shows  a  brief  iridescence  of  meaningless  changes 
and  bursts  into  dust. 

And  then  the  ruin  of  art  begins  to  infect  the  other  parts  of  mind. 
Thought  is  even  degraded  to  be  a  mere  scheme  of  symbols,  useful  for 
our  practical  life,  but  worthless  in  themselves.  We  are  shut  up  within 
the  impenetrable  wall  of  a  senseless  '  mind '  without  any  means  of  testing 
the  true  worth  of  the  bonds  that  seem  to  link  us  to  a  world  beyond, 
and  with  every  'reason'  to  suppose  them  illusory.  Morality  becomes 
the  preferences  of  a  social  group,  and  the  world-race  of  nations  is  then 
to  the  strongest,  who  grants  himself  all  the  sanctions  of  subjectivism. 
In  the  vast  struggle  of  these  years  we  are  not  striving  merely  to  make 
room  for  every  man's  illusions  against  one  tyrant's  ambitions.  That 
would  really  yield  the  senseless  chaos  that  freedom  seems  to  the  enemy 
to  be.  We  are  rather  labouring  to  fulfil  the  demands  for  obedience 
of  laws  that  claim  objective  worth  and  that  promise  to  give  our  souls 
the  dignity  of  reality,  of  being  members  in  a  coherent  intelligible  world. 

Another  source  of  subjectivism  was  created  by  the  critical  philosophy 
that  has  been  lauded  for  a  century  and  a  half  as  the  greatest  revelation 
achieved  by  thought.  It  is  true  we  must  examine  the  work  of  knowledge 
with  all  care  to  see  what  it  can  validly  establish.  But  if  our  conclusion 
be  that  true  knowledge  is  so  overlaid  with  the  devices  of  the  mind  itself 
that  it  is  impossible  to  say  anything  about  the  material  upon  which  it 
started  to  build,  we  are  driven  therewith  upon  the  rocks  of  subjectivism, 
no  matter  how  zealously  we  repeat  to  ourselves  :  "what  the  mind  once 
claimed  to  know  (really),  it  may  still  claim  to  know  (virtually)." 

Of  course  we  are  glad  to  have  cast  away  the  crude  illusions  of  know- 


216  THE  OBJECTIVITY  OF  BEAUTY  [ch. 

ledge  that  always  bind  our  thoughts  so  long  as  we  do  not  examine 
evidences  carefully  and  reason  with  caution.  In  a  sense,  however, 
these  illusions  are  only  the  natural  result  of  a  force  of  knowledge  that 
is  eager  to  work  and  has  not  yet  learnt  to  harness  itself  to  the  ideals 
of  completeness  and  security.  But  if  our  critical  study  of  knowledge 
is  to  lead  us  to  suppose  that  thought  itself  is  a  deceit  like  a  kaleidoscope, 
revealing  nothing  beyond  its  own  gyrations  but  the  existence  of  light 
itself,  then  have  we  cause  for  misery.  We  shall  be  prone  to  think  that 
the  older  'dogmatisms'  that  knew  nothing  of  critical  scepticism,  were 
really  sounder  products  of  the  spirit  than  the  critical  philosophy  itself. 
They  may  have  known  all  that  we  know  of  evidence,  consistency, 
and  system,  although  they  were  greatly  deficient  in  facts  and  left  in 
their  expositions  too  much  to  the  imagination  and  understanding  of 
their  readers. 

At  any  rate  a  sounder  vigour  is  beginning  to  show  itself  openly  in 
the  sciences  of  biology  and  in  philosophy.  We  now  see  that  chance 
of  itself  does  nothing,  and  that  selection  merely  selects  or  favours  one 
of  many  possibilities,  each  of  which  is  founded  upon  the  coherence  of 
objective  forces  that  we  call  laws  of  nature.  All  the  possibilities  of 
inanimate  or  animate  combination  are  realised,  if  only  the  forces  by 
which  they  are  surrounded  tolerate  them ;  and  they  endure  in  proportion 
to  their  stability  and  coherence.  Every  individual,  no  matter  how 
special  or  'rare'  he  be,  is  founded  upon  the  not  innumerable  forces 
that  make  an  infinite  variety  of  combinations  possible.  His  unique 
composition  involves  only  the  limited  gamut  of  the  world's  scale.  In 
respect  of  generality  and  immutability  the  underlying  laws  of  life  take 
rank  beside  even  the  laws  that  rule  the  great  bodies  of  the  universe. 
These  are  the  great  chimes  that  tell  the  years,  those  are  the  fleeting 
colours  of  an  orchestral  hour.  But  they  both  go  back  to  the  primary 
balance  and  beauty  of  pure  tone. 

Our  study  of  music  has  shown  us  that  order  and  systematic  relation- 
ship are  the  ground  and  basis  of  the  art,  not  capricious  selection  and 
convention.  We  have  indeed  to  learn  to  combine  the  fundamental 
forces  of  tone;  but  there  is  no  caprice  or  chance  in  the  forms  that 
please.  Chance  pertains  only  to  the  experiments  that  we  shall  at  any 
time  make,  within  a  certain  scope  allowed  by  the  age  in  which  we  live. 
We  cannot  yet  adopt  what  may  seem  beautiful  to  a  later  age.  Not  that 
we  may  not  already  see  some  beauty  in  it;  but  the  orderly  scheme  of 
beauty  we  have  thus  far  gathered  does  not  yet  offer  it  any  welcome. 


XXIII]  THE  OBJECTIVITY  OF  BEAUTY  217 

No  place  has  been  prepared  for  it  within  the  art.  Music  has  to  grow 
till  it  is  able  to  comprehend  each  special  beauty  as  its  own,  just  as  the 
mind  grows  gradually  towards  the  wisdom  of  its  years.  But  both  the 
beauty  and  the  wisdom  of  a  later  day  are  the  true  fulfilment  of  their 
earlier  forms,  children  of  the  same  elemental  order  and  purpose. 

It  is  evident  enough  that  there  is  a  structure  and  necessity  in  what 
we  find  to  be  beautiful.  That  is  true  in  broad  outline  at  least,  however 
difficult  it  still  may  be  to  carry  the  demonstration  down  to  the  finest 
details.  Beauty  is  of  our  creating  only  in  the  same  sense  that  the  forms 
of  life  are  the  result  of  selection.  It  comes  before  us  and  we  find  it  good. 
We  combine  its  simpler  parts  as  we  can,  and  we  then  find  laws  in  the 
conjunctions  that  please  us.  We  use  for  art  a  material  that  has  already 
been  wrought  from  the  ore,  as  it  were.  In  short  we  discover  beauty 
just  as  we  discover  truth  or  even  goodness. 

Beauty  is  not  beauty  merely  because  we  feel  pleasure  \nth  it.  It 
is  true  we  should  not  respond  to  it  or  feel  it  but  for  pleasure.  But  that 
is  really  a  tautology.  It  means  only  that  if  we  did  not  respond  to  it, 
we  should  indeed  not  do  so.  On  the  other  hand,  were  beauty  not  what 
it  is,  it  would  excite  no  feeling  in  us.  And  mere  feeUng  of  a  false  kind 
never  makes  beauty.  We  have  to  adjust  our  souls  so  as  to  feel  aright, 
just  as  we  have  to  be  careful  to  think  truly.  Feeling  is  merely  the 
thrill  that  beauty  strikes  through  our  being,  a  symbol  of  our  fitness 
to  share  its  high  functions.  We  are  aware  at  such  times  that  our 
experience  is  most  cogent  and  coherent,  as  it  is  likewise  in  true  thought 
and  deed. 

Moreover,  beauty  reaches  back  continuously  into  the  physical  world. 
The  tonal  volumes  and  pitches  that  are  our  own  experiences,  find  an 
objective  counterpart  or  parallel  in  the  excitations  spread  over  the 
sensitive  surface  of  the  ear.  And  the  simplest  parts  of  these  at  least 
depend  upon  equally  regular  processes  in  the  aerial  medium.  Although 
the  three  stages  of  the  physical,  physiological,  and  psychical  are  not 
precisely  alike  in  all  respects,  yet  their  differences  are  completely 
comprehensible  and  regular.  Except  in  so  far  as  we  cannot  yet  discern 
the  inherent  continuity  of  mind  and  matter,  there  is  no  mystery  in  their 
general  continuity  of  action,  as  far  as  hearing  at  least  is  concerned. 
What  comes  to  us  as  beauty  is,  therefore,  founded  upon  the  order  and 
regularity  of  the  natural  world.  In  fact  it  is  itself  a  part  of  the  natural 
world. 

If  anyone  cares  to  read  this  conclusion  as  "beauty  is  physical," 
no  great  objection  could  be  urged,  so  long  as  the  meaning  implied  is 


218  THE  OBJECTIVITY  OF  BEAUTY  [ch. 

not  altered  to  suggest  that  we  understand  the  continuity  running 
through  matter,  body,  and  this  'physical'  mind.  That  we  manifestly 
do  not  yet  understand.  If  our  comprehension  is  to  be  secured  by  setting 
down  the  realistic  constructions  of  ordinary  knowledge  and  of  science 
as  mere  classifications  of  phenomena  or  as  mere  symbols  or  counters 
for  thought,  we  are  surely  on  a  false  track.  No  one  is  really  inclined 
thus  to  sacrifice  the  results  of  science  as  illusory.  Nor  has  anyone 
succeeded  in  basing  what  we  know  in  science  in  the  broadest  sense  upon 
what  is  exclusively  phenomenal  to  us,  upon  sensations,  for  example, 
and  upon  nothing  else.  Nor  has  anyone  yet  shown  that  the  entities 
posited  by  science  are  essentially  of  the  nature  of  the  phenomenal 
units  of  our  sensory  experience  or  the  like.  What  the  elements  and 
complex  units  of  the  physical  world  really  are  in  themselves  apart 
from  their  order  and  regular  changes,  we  do  not  know.  If  we  pretend 
to  know  by  trying  to  believe  they  are  all  illusory  or  fictitious  ('reasonably 
fictitious,'  of  course),  we  only  deceive  ourselves.  For  we  can  never 
really  believe  that.  The  well  ordered  panorama, — all  of  which  to  the 
last  particle  we  can  see  with  direct  gaze, — that  some  would  have  us 
believe  in,  is  a  myth.  We  cannot  in  ourselves  be  all-seeing  and  all-wise. 
And  if  we  see  only  in  part,  and  that  with  great  labour  of  concentration, 
then  there  is  much  of  the  world  that  reveals  its  movements  and  forms 
to  us  only  through  our  own  very  limited  panorama  of  phenomena. 
Thus  there  must  be  many  things  in  the  world  whose  inner  being  remains 
indefinitely  hidden  from  us.  Beauty  may  be  a  part  of  the  physical 
world,  if  you  like,  continuous  in  action  with  it;  but  it  is  not  a  part  of 
that  real  world  which  we  call  physical  and  whose  inner  essences  are 
hidden  from  us.  For  we  have  the  full  being  of  beauty  before  us,  however 
difficult  it  may  be  for  us  to  understand  it  completely.  We  feel  it  entirely ; 
it  is  there  before  us  in  all  its  native  essence. 

If  beauty  is  thus  objective,  a  part  of  the  natural  world,  we  have  a 
greater  satisfaction.  What  we  have  before  us  is  the  promise  of  all. 
Beauty  then  belongs  essentially  to  the  world.  And  the  world  we  may 
look  upon  as  really  so  beautiful  as  we  naively  observe  it  to  be.  Its 
charms  are  not  the  mere  dreams  of  a  creature  of  chance,  dreams  that 
will  vanish  with  its  death.  We  may  be  sure  that,  wherever  there  is  a 
mind  to  view  the  sunset  or  to  hear  a  song,  provided  that  mind  has 
grown  to  the  stature  of  the  task,  there  beauty  will  be  evident.  There 
is  not  one  world  for  you  and  another  for  me,  but  one — according  to 
our  varying  powers  of  apprehension — for  all. 


xxm]  THE  OBJECTIVITY  OF  BEAUTY  219 

In  pre-critical  days  every  discovery  of  awakening  science  was  apt 
to  be  hailed  as  new  evidence  of  design  in  nature  and  of  the  presence 
cxf  God's  hand  in  the  world.  We  should  not  generally  now  be  so  simple 
as  to  imagine  that  law  and  order  form  a  sufficient  proof  of  dependence 
upon  an  omnipotent  person.  But  for  all  that  we  cannot  forgo  our 
wonder  and  admiration  at  the  awful  complexity  and  the  marvellous 
stability  of  things.  No  amount  of  belief  in  mechanism  will  render  us 
quite  cold  to  these  effects. 

We  may  indeed  tell  ourselves  that  our  minds,  having  been  evolved 
in  dread  of  nature's  rough  forces  and  of  the  savagery  of  beast  and  man, 
are  only  too  prone  to  be  moved  by  what  is  vast  and  great  and  complex. 
And  we  may  dispassionately  infer  that  our  notion  of  a  natural  body 
is  as  of  a  thing  we  might  ourselves  have  tried  to  make — a  machine 
and  its  works;  and  that  we  pass  therefrom  merely  by  association  of 
ideas  to  a  notion  of  the  effort  and  ingenuity  that  the  works  of  nature 
would  have  involved,  had  a  mind  like  ours  devised  them.  From  this 
point  of  view  we  read  the  effusions  of  the  earlier  philosophers  of  nature 
with  a  certain  amount  of  condescension. 

We  are  supported  in  this  attitude  by  the  reflexion  that  though 
mind  itself  may  feel  orderly  and  lawful,  such  a  feeling  is  but  illusion. 
Really  it  is  merely  subjective  and  accidental,  a  product  of  the  subtlest 
cerebral  chemistry  or  physics,  not  only  worthless  as  a  sign  of  manifest 
order,  but  obviously  quite  beyond  any  hope  or  possibility  of  analysis. 
We  can  take  as  our  test  of  mind's  action  no  formation  inherent  in  itself, 
no  continuity  of  process  that  binds  it  to  an  objective  world,  but  only 
the  success  of  its  functions,  which  in  its  turn  is  evidenced  only  by 
the  mind's  tolerance  of  them.  Not  only  can  the  existence  of  a  universal 
world  not  be  proved  by  mind,  but  we  must  finally  admit  that  whatever 
the  mind  is  content  to  accept  is  acceptable  (or  'true').  There  is  not 
only  no  permanent  world  from  which  the  world  might  seem  to  acquire 
order  by  merely  reflecting  it,  but  the  mind  itself  cannot  be  expected 
to  generate  order.  Its  autonomy  is  sheer  heteronomy,  its  sanctions 
are  its  own  acts,  its  only  law  is  mere  function. 

But  if  law  and  order  are  discovered  and  proved  for  mind  itself,  what 
then?  We  may  expect  to  hear  the  reply  :  what  you  have  shown  to  be 
so  orderly  is  not  mind  but  nature;  or,  you  have  merely  carried  through 
a  useful,  helpful  scheme,  which  has  no  further  validity  than  the  brevity 
and  expository  value  you  have  shown  it  to  possess.  Both  of  these 
interpretations  will  surely  in  the  end  prove  tiresome  and  nauseous. 
For  as  we  all  suppose  ourselves  to  have  some  sort  of  mind  of  our  own, 


220  THE  OBJECTIVITY  OF  BEAUTY  [ch.  xxiii 

we  shall  hardly  consent  to  abandon  it  all  to  nature;  nor  can  we  enjoy 
the  jest  of  a  pragmatic  world  for  ever.  We  shall  gradually  recognise 
that  the  order  of  as  much  '  mind '  as  we  at  any  time  understand  is  akin 
to  the  great  'design'  we  now  see  in  nature.  We  shall  become  convinced 
that  'mind'  is  of  a  piece  with  nature.  A  great  and  universal  objectivity 
runs  through  both. 

Shall  we  then  be  more  inclined  to  see  mind  in  nature, — a  vast 
thought,  actually  delineated  by  our  sciences,  'thinking'  itself  (or 
whatever  other  faculty  of  mind  we  may  conceive  it  to  be),  forwards 
by  its  rotations  and  syntheses  in  its  own  vast  sensorium  (or  conceptorium 
or  effectorium),  although  we  can  in  no  way  discern  the  quality  of 
consciousness  that  pervades  it?  Or  shall  we  humbly  infer  that  our  own 
minds  are  a  mere  mechanism  too,  rolling  on  because  it  once  got  started 
on  its  futile  gyrations,  endowed  with  a  stability  that  is  a  synonym 
for  'it  is  what  it  is'?  Shall  we  not  rather  learn  to  see  that,  in  mind, 
not  only  is  the  essence  of  order  and  of  law  actually  made  manifest,  but 
inner  coherence  and  stability  has  even  become  the  ideal  of  all  its  effort 
(pervading  the  great  faculties  of  sense-beauty,  concept-truth,  and 
emotion- joy,  love,  goodness)?  When  these  ideals  seem  attainable,  its 
very  being  is  rapture;  it  is  misery  so  long  as  they  seem  indefinitely 
elusive. 

Whatever  our  general  argument  or  our  detail  of  knowledge  may  be, 
we  shall  probably  all  incline  always  to  some  sort  of  an  idealism,  however 
much  of  reality  we  may  at  the  same  time  find  around  us  whose  kinship 
of  being  with  us  we  cannot  fully  prove,  at  least  as  regards  the  last 
essential  phase  of  that  proof.  And  we  may  do  well  to  be  so  inclined. 
It  is  not  to  be  expected  that  a  creature  of  this  world  should  be  willing 
to  sacrifice  the  only  direct  vision  of  being  it  possesses  for  any  of  the 
forms  of  other  beings  that  its  knowledge  may  describe.  Until  we  learn 
that  mere  form  can  constitute  a  being,  we  shall  hope  in  our  modesty 
that  the  things  of  this  world  are  essentially  (or  in  their  inherent  being) 
greater  (spirits)  than  we  are,  and  in  our  pride  that  they  have  in  themselves 
some  portion  of  the  insight  of  mind  and  some  glow  of  the  emotion  we 
find  in  ourselves. 

The  knowledge  we  have  won  may  at  least  make  us  hope  and  believe 
that  in  beauty  we  are  in  direct  touch  with  a  real  aspect  of  the  world. 
And  if  the  world  appears  to  us  in  the  order  of  law  and  in  the  harmony 
of  beauty,  why  not  also  in  the  kingdom  of  goodness  and  love? 


CHAPTER  XXIV 

AESTHETICS  AS  A  PURE  SCIENCE 

Aesthetics  as  a  pure  science  is  simply  a  part  of  the  pure  science  of 
psychology.  So  all  the  characteristic  features  of  the  latter  notion 
appertain  to  the  former. 

Psychology  is  concerned  with  the  parts  and  the  structure  of  what 
is  commonly  called  'mind.'  That  includes,  according  to  the  doctrines 
of  modern  science,  not  only  such  things  as  memory,  thought,  and 
emotion,  not  only  the  bodily  feelings  of  hunger,  pain,  warmth,  posture, 
and  movement,  but  also  all  sights  and  sounds,  distances,  forms  and 
spaces.  Of  course  some  folks  object  to  having  these  last  things  called 
sensations  and  experiences.  But  that  does  not  matter  at  all  for  the 
present,  so  long  as  we  have  good  reason  to  group  them  together  and  with 
the  bodily  feelings  just  mentioned.  All  these  things  call  for  systematic 
study  together. 

In  fact,  they  seem  to  form  a  distinct  world  by  themselves  and  apart 
from  all  that  science  can  tell  us  of  the  physical  forces  that  arouse  pain, 
warmth,  sound,  sight,  or  of  physical  form  and  space.  The  ordinary 
man  lives  in  his  world  of  sensory  feeling  as  he  finds  it;  he  does  not 
concern  himself  with  the  physics  of  sound,  light,  or  warmth,  or  even 
of  space.  Psychology  is  the  name  of  the  science  that  has  to  arrange 
and  to  explain  the  parts  of  this  world  in  relation  to  one  another,  however 
it  may  afterwards  (or  otherwise)  join  hands  with  any  other  science 
(say  physiology  or  physics)  to  show  how  feelings  depend  upon  the  body 
or  upon  matter.  This  systematic  work  of  psychology  has  to  be  done 
without  filling  up  gaps  of  ignorance  by  disquisitions  on  the  relation  of 
the  feelings  to  the  body  or  to  matter,  and  without  giving  theories  about 
feelings  that  are  merely  inferences  from  what  we  know  of  matter  or  of 
body  (unless  the  temporary  deficiency  of  our  knowledge  makes  such 
an  effort  inevitable).  That  is  what  is  meant  by  speaking  of  fure 
psychology. 

The  work  of  pure  aesthetics  is  similar — to  give  a  full  description, 
systematic  arrangement,  and  so  an  explanation  of  every  work  of  art 
in  so  far  as  it  is  directly  before  the  spectator's  (or  artist's)  mind  without 
any  regard  for  all  the  (otherwise  so  important)  facts  and  questions 


222  AESTHETICS  AS  A  PURE  SCIENCE  [ch. 

relating  to  the  work  of  art  as  a  physical  thing  (marble,  canvas,  paint,  etc.) 
or  to  its  effect  upon  the  eye,  the  ear,  the  nerves,  the  brain. 

It  has  taken  science  very  long  to  find  out  that  the  brain  is  the  organ 
of  mind,  so  to  speak.  Most  people  have  still  only  the  vaguest  idea  of 
the  connexion  between  the  two.  But  this  ignorance  in  no  way  alters 
the  efficiency  of  their  minds.  They  can  observe  their  impressions  and 
the  workings  of  their  minds  quite  clearly;  they  know  whether  they 
feel  pleased  or  displeased,  with  reason  or  for  some  motive,  whether 
they  are  sure  of  their  conclusions  or  uncertain,  whether  their  conduct 
seems  to  them  good,  indifferent,  or  disgraceful  enough  to  need  conceal- 
ment. Of  course,  they  do  not  therefore  know  everything  about  their 
minds;  they  may  deceive  themselves  in  feeling,  knowledge,  and 
conscience ;  but  yet  we  all  know  and  believe  that  if  they  set  their  minds 
more  vigorously  and  honestly  to  work,  they  might  improve  its  action 
indefinitely.  In  the  same  way  the  finest  art  was  in  existence  long  before 
men  knew  precisely  of  what  marble  and  paint  consist  or  how  musical 
instruments  produce  their  tones.  A  composer  accepts  or  rejects  a  tone 
for  a  melody  or  a  chord  for  a  progression,  not  because  he  knows  some- 
thing of  its  physical  nature,  not  even  because  he  knows  what  partial 
tones  it  contains  or  implies,  but  merely  because  it  is  beautiful  and  fits 
beautifully  into  the  work  at  the  point  in  question.  He  has  all  the 
material  directly  before  his  mind  by  which  to  judge  of  this  without 
needing  any  scientific  knowledge  at  all. 

It  is  the  task  of  a  pure  psychology  in  general  and  of  aesthetics  in 
particular  to  give  as  connected  an  account  as  possible  of  all  that  thus 
goes  on  before  and  in  the  mind. 

There  are  many,  however,  who  say  that  though  much  can  be  found 
out  about  the  things  of  the  mind,  the  account  can  never  be  made 
complete  or  closed.  That  we  shall  see  in  the  course  of  time,  if  we  try. 
We  have  at  least  a  great  promise  of  approximate  completeness;  for 
do  we  not  expect  the  work  of  the  mind  and  of  art  to  grow  more  and 
more  consistent  and  perfect?  So,  to  begin  with,  we  have  perhaps  even 
more  cause  to  believe  in  the  uniformity  and  inherent  consistency  of 
mind  than  of  nature.  Whether  mind  is  in  some  region  or  process  chaotic, 
we  shall  have  cause  to  believe  if  we  fail  utterly  to  comprehend  some 
portion  of  its  ways. 

But  there  are  mysteries  both  in  matter  and  in  mind.  The  darkest 
mystery  of  the  physical  world  is  why  or  how  it  ever  was  created  and 
set  agoing.    The  very  question  is  a  sort  of  nightmare,  through  which 


XXIV]  AESTHETICS  AS  A  PURE  SCIENCE  223 

we  help  ourselves  by  the  recital  of  beautiful  myths.  The  problem  of 
the  origin  of  mind  is  just  as  baflOling.  Science  evades  these  ultimate 
issues.  Its  only  concern  is  to  reduce  the  world  or  mind  to  its  simplest 
foundations.  But,  given  these  and  what  it  learns  of  their  nature  in  the 
course  of  analysis,  it  hopes  to  show  how  the  complex  world  and  the 
mind  we  know  result  therefrom,  without  the  assumption  of  any  inter- 
ference 'from  without,'  We  do  not  now  suppose  that  God  steps  in,  as 
occasion  demands,  to  wind  up  the  clock  of  the  physical  universe  or  to 
mend  its  works.  Nor  need  we  suppose,  once  the  elements  of  mind 
have  been  given  by  the  brain  it  is  dependent  upon,  and  the  process  of 
complication  and  interaction  of  these  elements  has  begun,  that  the 
brain  here  and  there  exerts  its  influence  anew  to  keep  the  process  going. 

This  last  idea  may  be  illustrated  in  a  crude  way  by  reference  to  the 
problem  of  consonance.  Instead  of  supposing  that  there  is  something 
about  the  mass  of  sound  constituted  by  two  tones  an  octave  apart 
or  by  the  common  major  chord  that  we  can  discover,  describe,  and  set 
up  as  the  object  or  cause  of  the  pleasant  feeUng  we  have  in  connexion 
with  it,  some  folks  speak  as  if  we  heard  the  two  or  three  tones  precisely 
as  we  hear  each  of  them  separately  (only,  of  course,  now  together), 
as  if  there  were  equally  little  or  nothing  in  either  case  to  account  for 
any  unpleasantness,  and  as  if  the  brain,  so  to  speak,  wired  a  separate 
message  to  the  mind  in  the  code- word  'pleasant  feeling,'  which  being 
somehow  decoded  by  the  mind  would  mean  :  "  the  tone-group  dispatched 
by  separate  route  is  to  be  'pleasant,'  i.e.  runs  nicely  through  my  nerve 
centres." 

If  this  sort  of  thing  were  true,  the  only  problem  of  psychology 
would  be  to  make  a  rapid  catalogue  of  the  brain's  code-words,  and  to 
try  by  all  sorts  of  experiments  to  catch  the  brain  in  the  very  act  of 
sending  off  a  code- word  and  to  see  what  was  going  on  in  it  at  the  moment 
(that  was  nice  or  nasty).  And  as  a  matter  of  fact  that  is  exactly  what 
the  program  of  psychology  involves  according  to  the  present  notions 
of  (probably)  the  majority  of  its  professed  exponents. 

Whatever  limits  lurk  hidden  in  the  notion  of  pure  psychology  or 
aesthetics,  that  kind  of  idea  must  surely  be  a  travesty  of  mind  and 
art  as  we  know,  or,  at  least,  'feel,'  them.  It  is  absurd  to  suppose  that 
our  minds  are  the  mere  puppets  of  a  brain-and-matter  show.  If  that  is 
the  sort  of  world  we  live  in,  the  sooner  the  silly  play  is  over  the  better. 
The  opposite  extreme  is  far  more  worthy  of  the  vast  and  orderly  cosmos 
we  know  the  world  to  be,  at  least  the  world  of  nature.  Let  us  suppose 
that  in  the  mind  we  shall  find  the  greatest  inner  order,  coherence, 


224  AESTHETICS  AS  A  PURE  SCIENCE  [ch. 

self-sufl&ciency,  completeness  that  is  anywhere  detectable  at  all.  Let 
us  never  rest  till  we  have  gone  far  towards  realising  this  ideal  by  the 
most  stringent  methods  of  science.  And  let  us  then  learn  to  see  the 
great  world  in  the  light  of  the  little  mind  we  know,  permeated  and 
bound  fast  by  inner  forces  like  those  that  call  us  forwards, — truth, 
beauty,  goodness, — although  we  know  the  stability  of  these  cosmic 
aspirations  only  in  their  outer  forms,  not  at  all  in  their  inherent  essence 
or  'spirit.' 

The  pure  psychology  that  has  been  expounded  in  the  preceding 
chapters  is  at  the  same  time,  in  so  far  as  we  have  confined  our  attention 
to  the  foundations  of  music,  a  pure  aesthetics.  We  have  found  a  basis 
for  the  art  that  at  no  point  transgresses  the  range  of  what  is  before 
the  mind  of  the  creative  or  appreciative  musician.  No  gap  in  our 
knowledge  of  the  foundations  of  music,  as  we  hear  and  enjoy  it,  has 
been  filled  by  our  knowledge  of  the  physical  processes  of  sound  or  of  the 
bodily  functions  of  hearing. 

Of  course  the  results  attained  are  in  considerable  part  new;  they 
do  not  embody  at  all  what  the  musician  may  himself  have  thought  of 
music.  But  that  is  no  real  difficulty.  Many  an  appreciative  listener 
enjoys  a  new  work  of  music  without  being  able  to  analyse  it  or  to  say 
of  what  parts  it  consists;  and  yet  no  one  would  suggest  that  the  work 
was  not  completely  before,  or  accessible  to,  his  mind.  In  the  knowledge 
of  a  thing  there  are  two  important  parts — the  knowledge  and  the  thing. 
And  two  relations  of  time  are  possible  between  them  :  they  may  come 
into  existence  practically  at  the  same  time,  or  the  thing  may  precede 
the  knowledge  by  an  indefinite  period.  This  latter  alternative  holds 
not  only  for  the  objects  of  nature,  but  in  some  cases  for  the  things 
of  the  mind  as  well.  It  is  a  familiar  fact  that  various  difference-tones 
have  been  discovered  only  comparatively  recently;  and,  before  the 
time  of  Tartini  or  thereby,  these  were  quite  unknown.  Similarly  partial 
tones  were  not  heard  and  known,  but  only  heard,  to  be  in  timbre  before 
Helmholtz's  time^.  And  if  my  exposition  is  on  the  right  lines,  I  may 
claim — to  the  best  of  my  knowledge  and  belief — that  no  one  knew 

^  In  this  sentence  special  emphasis  should  be  laid  upon  the  word  timbre.  Gevaert 
(16,  118  f..  163 1.)  is  convinced  that  the  existence  of  the  first  harmonic  was  known  to  Aristotle. 
Cf.  Gevaert's  translation  (16,  i9):  "Pourquoi  dans  la  consonance  d'octave  le  grave  repro- 
duit-il  I'intonation  de  I'aigu,  tandis  que  I'aigu  ne  reproduit  pas  I'intonation  du  grave?.... 
Et  (en  effet)  la  corde  hypate,  au  moyen  de  sa  division  (en  deux  parties  egales),  produit 
deux  netes  distinctes."  "Does  not  Aristotle  in  his  treatise  on  the  Soul  say:  17  ^wv^  ffvfjL<f>u)vla 
tIs  iariv  (tone  is  a  sort  of  chord),"  (16,  ii9)? 


XXIV]  AESTHETICS  AS  A  PURE  SCIENCE  225 

before  what  tone  and  interval  in  their  various  aspects  really  are,  although 
every  musical  mind  must  have  felt  what  they  are  a  countless  number 
of  times  previously. 

Our  study  of  the  foundations  of  music  has  now  brought  us  clearly 
within  the  range  of  the  musician's  present  interests.  It  is  unnecessary 
for  him  to  know  precisely  the  nature  of  tone  or  of  interval  in  their 
simpler  aspects,  or  to  understand  the  nature  of  the  blend  of  tone  (timbre). 
But  no  musician  can  produce  good  work  without  knowing,  as  well  as 
feeling,  the  effects  of  the  various  intervals  upon  the  flow  of  his  melodies 
or  harmonies.  The  analysis  of  these  things  is  the  lore  of  the  music 
student  that  we  have  freely  used  as  a  basis  of  induction.  By  connecting 
the  results  of  induction  with  the  psychology  of  tone  and  interval  in 
their  simplest  aspects  we  have  made  it  possible  for  the  musician  to 
follow  the  growth  of  music  from  its  foundations  in  mere  tone  up  to  the 
strands  and  links  that  bind  tones  into  coherent  music.  The  study  can 
now  be  taken  over  by  the  musician  himself  and  pursued  without  the 
further  help  of  psychology,  in  so  far  as  this  is  the  general  and  systematic 
study  of  all  kinds  of  sensations  and  experiences. 

No  doubt  there  is  much  still  to  be  done,  the  results  of  which  may 
reflect  greatly  upon  the  disposition  of  the  ground  thus  brought  under 
control.  But  the  musician  should  no  longer  feel  that  he  has  no  firm 
place  upon  which  to  build  the  systematic  structures  that  seem  to  be 
needed  for  the  housing  of  his  musical  analyses.  We  have  now  a  clearer 
view  of  what  music  really  is;  and  we  can  venture  to  say  that  we  now 
understand  much  of  what  heretofore  seemed  to  be  so  mysterious  in 
the  art.  In  the  following  we  shall  review  a  few  of  these  old  problems 
of  music  that  are  at  the  same  time  special  forms  of  the  general  problems 
of  aesthetics. 

There  is  a  special  ease  and  suggestiveness  in  claiming  for  music 
the  need  for  a  pure  aesthetics.  It  has  been  asserted  in  a  much  quoted 
sentence  that  "all  art  tends  to  the  condition  of  music"  (quoted  from 
33,  227).  By  this  is  meant  that  music  is  an  art  that  is  obviously  pure 
and  disinterested,  and  that  the  other  arts  can  only  with  difficulty 
reach  the  level  of  purity  that  is  natural  and  almost  inevitable  in  music. 
But  all  arts  are  inherently  pure;  they  all  are  only  'for  art's  sake.' 

Probably  the  chief  element  in  this  distinction  is  the  fact  that,  while 
it  is  very  difl&cult  to  exclude  from  the  arts  of  sight  the  representative 
functions  of  vision,  there  are  so  few  representative  processes  in  tonal 
sound  that  the  arts  of  sound  remain  wholly  unrepresentative.   How  few 

w.  p.  M  16 


226  AESTHETICS  AS  A  PURE  SCIENCE  [oh. 

of  the  things  around  us  make  a  tuneful  noise!  Writers  on  the  theory 
of  program  music  (e.g.  39;  cf.  42)  have  been  able  to  compile  only  a 
brief  list :  the  cuckoo  and  the  cock  are  alone  unmistakeable ;  the 
nightingale,  the  thrush,  and  some  others  sing  in  a  way  that  can  be 
imitated  and  recognised  with  fair  ease;  but  the  sounds  of  most  things 
are  really  noises  that  are  so  hard  to  imitate  on  musical  instruments  that 
their  recognition  cannot  be  assured.  Besides,  these  representative 
sounds  offer  so  little  scope  for  artistic  modification  and  construction 
that  their  introduction  into  a  musical  work  is  obviously  inappropriate 
and  produces  a  ludicrous  effect. 

But  there  is  an  aspect  of  the  question  that  seems  to  be  of  much  more 
general  importance  than  these.  Sounds  cannot  really  be  said  to  represent 
the  animal  or  object  that  emits  them,  any  more  than  a  cloud  of  dust 
represents  a  motor-car,  or  a  trail  of  steam  a  railway  engine.  The  form 
of  any  object  represents  it  or  is  it  in  an  intimate  way  that  characterises 
no  other  effect  produced  by  it.  For  in  all  these  the  form  presented 
in  the  effects  differs  entirely  from  the  form  of  the  object  itself.  In  the 
case  of  sound  this  is  especially  so.  The  formal  aspects  of  sound  (tone, 
volume,  and  interval),  and  the  lines  of  melodies  have  no  relation  at  all 
to  the  shape  of  the  objects  that  emit  them.  They  cannot  then  properly 
represent  them.  But  the  pictorial  representation  of  a  house  or  of  a 
man  does  not  merely  make  us  think  of  the  house  or  the  man;  it  is 
(pictorially)  the  house  or  the  man. 

It  is  in  no  way  the  merit  of  music  that  it  excludes  representation. 
It  does  so  merely  because  few  things  produce  more  than  a  characteristic 
noise  or  timbre,  and  those  features  of  sounds  are  the  hardest  of  all  to 
reproduce  exactly.  In  short  music  cannot  be  representative.  Pictorial 
art,  on  the  contrary,  cannot  readily  exclude  the  representative  aspects 
of  things  without  sacrificing  much  that  is  of  the  highest  interest  and 
so  impoverishing  itself  greatly.  It  can,  of  course,  limit  its  scope  to  the 
purely  ornamental;  but  valuable  and  beautiful  as  that  is,  our  interest 
in  representative  forms  is  so  intense  that  we  constantly  long  to  see  the 
forms  of  pure  ornament  vivified  by  the  use  of  representation. 

There  is,  therefore,  a  comparatively  limited  field  for  the  art  of  sound 
to  build  upon,  great  and  beautiful  as  are  the  edifices  it  raises  thereon. 
Between  this  field  and  the  emotional  life  that  is  expressed  in  the  art 
there  is  a  direct  and  continuous  transition.  The  claim  of  music  to  be 
the  standard  of  all  art  implies  that  in  the  pictorial  arts  all  the  ranges 
of  experience  other  than  those  of  colour  and  form  that  enter  into  artistic 
appreciation  are  unlawful  intruders.    Their  presence  is,  as  it  were,  a 


XXIV]  AESTHETICS  AS  A  PURE  SCIENCE  227 

concession  to  our  prevailing  habits  of  mind,  not  a  proper  part  of  the 
art's  action.  The  probability  of  this  being  true  is,  to  say  the  least, 
extremely  small. 

For  it  is  the  peculiarity  of  the  aesthetic  attitude  that  it  is  always 
direct;  it  appreciates  an  aesthetic  object  only  in  so  far  as  it  is  before 
the  mind,  not  at  all  in  so  far  as  it  is  representative.  Of  course,  representa- 
tion is  not  excluded;  but  the  material  thus  brought  before  the  mind 
is  of  value  only  in  so  far  as  it  is  what  it  is,  not  in  so  far  as  its  purely 
representative  function  is  concerned.  The  primary  basis  of  music  is 
undeniably  the  tonal  material  and  all  its  forms  and  movements.  It  is 
not  easy  to  make  these  representative  without  effort;  but  beautiful 
music  might  conceivably  please  merely  because  it  reminded  the  listener 
of  some  beloved  person  who  once  made  such  music  or  because  it  indicated 
much  skill  and  energy  in  the  player.  Many  concert-goers  are  inclined 
to  judge  musical  performances  from  the  point  of  view  of  their  acrobatic 
perfection  and  virtuosity,  if  not  even  from  that  of  the  loudness  of  sound 
attained.  In  the  same  way  the  mere  colours  and  forms  of  ornament 
may  suggest  many  extraneous  ideas  to  the  spectator.  But  neither  of 
these  deviations  from  the  artistic  attitude  make  us  desire  to  get  rid 
of  the  suggestive  sensory  material  from  the  arts.  It  is  not  forbidden 
that  a  piece  of  music  shall  be  difficult  to  perform,  but  only  that  it  shall 
not  be  performed  so  that  its  technical  accomplishment  stands  forth  as 
the  chief  object  of  interest  to  the  discredit  of  the  inherent  beauty  of 
all  its  sounds.  The  patchy  bedabbled  canvasses  that  are  often  seen  in 
exhibitions  err  in  a  similar  way  by  making  the  method  of  work  so 
obvious  that  the  eye  is  distracted  from  the  intended  beauty.  But  no 
one  would  desire  that  the  method  of  work  should  always  be  so  hidden 
away  as  to  be  undetectable,  merely  because  there  are  many  who  might 
rather  observe  it  than  the  picture's  beauty.  Why  then  should  we  wish 
to  banish  the  representative  functions  from  visual  arts  because  some 
folke  may  take  the  work  as  an  illustration  and  not  as  art? 

It  is  surely  a  great  mistake  to  try  to  limit  the  basis  of  construction 
of  any  art  to  a  certain  range  of  'experience,'  even  if  it  be  to  the  sensory 
range.  On  what  grounds  shall  we  fix  our  limits?  Take,  for  instance, 
the  space  of  vision  that  is  so  important  even  in  the  flat  arts  of  sight. 
A  solid  sculpture  seems  to  be  the  only  solid  type  of  artistic  object. 
A  picture  cannot  really  be  solid;  its  space  is  only  indicated;  if  it  were 
not  so,  we  should  take  it  for  a  solid  'panorama,'  as  we  see  a  room 
reflected  in  a  mirror.  The  picture's  space  is  only  perspectival  and 
'tonal,' — in  both  forms  'suggested'  to  the  eye,  not  given.   But  space  is 

15—2 


22S  AESTHETICS  AS  A  PURE  SCIENCE  [ch. 

one  of  the  most  powerful  interests  of  pictorial  art, — atmosphere, 
distance,  roominess,  etc.  Is  it  for  that  reason  a  foreign  element  in  the 
art?  Certainly  not :  for  the  simple  reason  that  the  spaces  of  art  are 
beautiful, — as  beautiful  as  are  the  lines,  areas,  and  forms.  If  we  can  look 
upon  the  latter  without  relation  to  the  real  distances  and  shapes  of 
physical  things  that  they  may  resemble,  we  can  just  as  readily  survey 
the  space  of  a  picture  without  relation  to  the  cubic  yards  and  miles 
of  physical  space  to  which  it  may  be  equivalent. 

And,  then,  what  sort  of  an  object  for  aesthetic  contemplation  would 
a  solid  sculpture  or  bronze  be  that  had  no  resemblance  to  any  known 
solid  object,  that,  in  other  words,  was  devoid  of  all  representative 
force?  If  it  were  not  purely  ornamental,  it  would  tend  to  lack  the 
coherence  of  design.  If  an  architecture  is  not  merely  a  pretty  symmetrical 
surface,  it  must  be  welded  upon  the  design  of  the  building  it  covers, 
so  that  it  expresses  its  functions  beautifully,  as  a  beautiful  body  shows 
the  graces  of  the  human  frame.  It  is  a  distinctly  inferior  art  that 
ignores  the  internal  body  and  merely  conceals  it  within  a  superficies 
of  beauty.  Why  should  not  doors  and  windows  and  turrets  be  moulded 
upon  a  rocky  hillside  instead?  But  if  they  belong  sensibly  only  to  the 
cathedral  or  to  the  house,  we  must  each  time  make  an  imperishable 
unity  in  beauty  of  the  inner  demands  and  the  outer  show.  The  designs 
of  solid  art  thus  inevitably  find  their  springs  in  the  designs  our  furniture, 
our  houses,  our  bodies  and  our  clothes  require.  Where  then  is  the 
artistic  irrelevance  of  such  things  of  the  world?  How  can  we  desire  to 
exclude  them  from  our  art  and  to  make  its  only  content  the  beauty 
of  ornament?  It  is  not  given  to  us  to  carve  the  mountains.  But  when 
they  reveal  their  design  in  beautiful  forms,  awesome  and  uplifting,  are 
they  too  not  works  of  art,  even  though  the  artist  be  nature  herself? 

The  only  escape  from  this  conclusion  involves  the  acceptance  of  an 
essential  irrelevance  in  all  art,  namely  that  what  is  beautiful  is  not  what 
is  before  us  in  the  aesthetic  object,  but  only  the  fact  that  that  object 
is  the  outcome  of  the  artist's  desire  to  express  himself.  That  is  out  and 
out  heteronomy.  The  only  beautiful  way  in  which  a  person  can  express 
himself  is  obviously  to  express  himself  in  himself.  Then  he  and  his 
expression  are  one  in  perfect  coincidence,  and  beautiful.  But  when 
a  man  makes  a  work  of  art,  he  makes  an  object  that  expresses  itself 
as  independently  of  him  thereafter  as  his  grown  son  ever  could.  In 
the  ideal  creation  the  artist's  personaHty  would  be  as  completely 
indiscernible  as  is  the  hand  of  God  in  nature.  The  works  of  such  a 
man  would  really  create  themselves;  they  would  spring  into  being  in 


XXIV]  AESTHETICS  AS  A  PURE  SCIENCE  229 

their  fundamental  nucleus  of  purpose  or  design,  and  they  would  clothe 
themselves  merely  by  the  unfolding  and  complication  of  that  first 
germ.  We  may  well  believe  from  many  indications  that  the  greatest 
works  of  art  have  thus  come  into  being.  The  greatest  artist  in  his  greatest 
moments  seems  not  to  mould  and  to  form  his  works  but  merely  to  yield 
himself  to  the  impulses  of  artistic  force.  He  is  not  so  much  a  maker  as 
a  discoverer  of  beauty,  however  much  he  may  have  to  grope  and  to 
search  before  he  finds  the  true  beauty.  Its  truth  has  no  relation  to  the 
length  or  manner  of  his  search.  His  sole  task  is  by  some  means  or  other 
to  find  the  true  beauty  and  to  recognise  it  then. 

When  the  older  writers  said  that  art's  function  was  to  imitate  nature, 
they  may  not  have  been  so  far  from  art's  secret  as  we  are  now  usually 
inclined  to  think.  We  know,  of  course,  that  the  beautiful  Venus  is  not 
found  amongst  living  women  and  then  merely  portrayed.  But  we  must 
not  forget  that  she  has  the  whole  design  of  woman,  that  she  expresses 
the  ideal  of  woman  as  she  lives  and  moves,  and  that  her  great  value 
for  us  lies  in  that,  as  well  as  in  her  artistic  embodiment  in  stone.  In 
the  Venus  we  do,  in  a  sense,  really  'imitate'  nature;  we  take  her  design 
and  the  surface  she  has  given  it  and  in  the  whole  we  wrap  up  and 
preserve  for  our  souls  the  elusive  perfection.  Would  it  be  the  same  if 
we  made  an  equally  beautiful  work  of  an  organic  type  that  has  no 
counterpart  or  kin  in  nature?  We  not  only  lack  the  underlying  design, 
but  we  do  not  even  feel  the  need  for  its  existence.  When  the  need 
arises,  the  design  soon  follows  and  grows;  and  in  time,  with  the  fullness 
of  experience  and  understanding,  its  own  beauty  permeates  the  whole. 

Of  course  we  may  want  to  assert  that  we  thus  better  nature  rather 
than  imitate  her.  There  is  no  harm  in  this  preference,  provided  we  do 
not  in  our  conceit  imagine  that  even  in  our  highest  conceptions  we  have 
created  much.  We  have  been  given  almost  all  we  have,  even  if  we  have 
added  our  labour  to  it.  It  is  really  absurd  nowadays  to  keep  up  the 
foohsh  belief  that  nature  is  never  beautiful,  but  only  the  compositions 
of  man.  On  the  contrary,  we  now  feel  that  man  is  the  child  of  nature, 
and  that  in  our  best  moments  we  do  but  see  more  clearly  what  nature 
means  with  us  in  all  her  doings.  It  is  not  art's  task,  to  be  sure,  merely 
to  illustrate  nature.  But  there  could  be  no  greater  task  for  the  visual 
arts  than  to  discover  nature's  inner  designs  and  to  reveal  the  full 
harmony  of  these  with  the  forms  and  actions  into  which  she  weaves  them. 

It  is  not  given  to  music  to  reveal  the  ideals  of  our  real  world.  But, 
using  a  Hegelian  inversion,  we  might  well  claim  that  music's  task  is 


230  AESTHETICS  AS  A  PURE  SCIENCE  [ch. 

to  reveal  the  realities  of  our  ideal  world.  It  has  often  been  urged  that 
music  expresses  much  more  intimately  and  purely  the  life  of  man's 
soul  than  any  other  art.  How  it  does  this  may  be  inferred  with  some 
probability  of  correctness  from  the  results  of  our  analysis  of  music. 
We  have  learned  that  music  is  essentially  paraphony,  an  interweaving 
of  simultaneous  melodic  rivulets.  Rhythm  is  a  most  important  element 
in  the  art,  though  it  has  not  been  the  object  of  our  investigation  at  all. 
The  motional  connexions  of  rhythm  are  very  powerful  and  combine 
with  those  of  melody  to  form  a  more  cogent  whole  than  either  could 
supply  alone.  The  variable  features  of  the  paraphony  and  of  the  rhythm 
of  music  are,  therefore,  thoroughly  motional.  They  may  vary  in  speed 
and  figure  (rhythm  in  the  special  sense).  Music  may  speak  in  one  voice 
or  in  two  or  more.  These  may  speak  a  common  language,  or  they  may 
utter  different  thoughts.  Or,  when  the  articulation  of  all  the  voices, 
or  of  all  but  one,  is  reduced  in  favour  of  their  common  sentiment,  an 
atmosphere  of  harmony  or  discord  in  all  its  colours  and  changes  may  be 
portrayed.  Loudness  will  indicate  its  strength,  gentleness  its  peace, 
height  of  pitch  its  brightness  or  gaiety,  depth  its  sadness  or  gloom. 

With  all  these  means  of  variation  it  is  only  necessary  to  bring  the 
motions  of  music  into  some  sort  of  correspondence  with  the  character 
of  the  acts  and  energies  of  man  for  it  to  be  able  to  express  his  soul's 
life.  Fast  or  slow,  vigorous  or  reposeful,  sombre  or  gay,  single  or  dis- 
tressed, loquacious  or  reflective,  clear  or  suspended  in  doubt,  no  mood 
can  occur  that  cannot  be  depicted  in  its  general  character  and  course. 
And  yet  there  is  withal  no  representation  in  the  art  thus  far. 

Nor  is  there  any  symbolism  or  convention.  We  do  not  agree,  as  it 
were,  to  look  upon  the  activities  of  a  musical  work  as  symbols  of  our 
soul's  life.  They  are  merely  what  they  are, — the  motions  of  melodies 
and  their  tonal  conjunctions  and  changes.  We  enjoy  them  in  the  first 
place  because  these  things  are  directly  enjoyable  by  us  or  beautiful  in 
themselves.  But  they  gain  in  interest  and  passion  by  the  fact  of  their 
natural  resemblance  to  the  'movements'  and  activities  of  our  soul  as 
a  whole,  without  our  necessarily  thereby  thinking  of  our  soul  at  all. 

We  touch  here  on  a  general  problem  of  aesthetics  that  has  often 
seemed  very  mysterious  indeed.  It  is  the  problem  of  introjection,  the 
projecting  into  the  external  phenomena  of  the  senses  of  moods  and 
sentiments  that  are  known  only  in  so  far  as  they  occur  in  ourselves. 
Why  does  a  fa9ade  or  a  trellis  look  excited  or  calm?  How  can  the  sky 
look  angry,  or  the  sunset  full  of  promise  and  hope?  Are  we  not  forced 
to  believe  that  only  some  obscure  analogy  or  association  brings  these 


XXIV]  AESTHETICS  AS  A  PURE  SCIENCE  231 

notions  to  our  minds  in  contemplating  the  object  and  that  we  then 
think  the  object  looks  the  character  that  it  merely  makes  us  think  of? 

We  may  perhaps  draw  a  better  conclusion  from  our  own  case  of 
music.  Music  not  only  seems  restless  upon  occasion,  it  really  is  so. 
Now  it  moves  at  one  pace,  now  at  another.  And  the  motional  connexions 
of  melody  are  as  much  motions  in  their  own  way  as  visual  motions  are 
in  theirs.  They  are  tonal  motions  :  not  spatial,  it  is  true,  as  visual 
motions  are,  but,  apart  from  this  difference  of  cognitive  reference, 
there  is  no  great  phenomenal  difference.  And  art  is  concerned  only 
with  the  phenomenal;  it  is  not  a  practical  or  a  scientific  issue. 

Again,  music  not  only  seems  single  or  involved,  it  is  so,  quite  as 
much  as  a  train  of  thought  or  of  sentiment  can  be  single  or  involved. 
Sentiments  and  tones  can  both  unite  into  one  functional  series,  giving 
an  effect  of  arrest  or  of  satisfaction  (of  tone  or  of  sentiment);  or  they 
may  interlace  in  disagreement,  producing  a  confusion  that  prompts 
us  to  seek  their  (re-)solution.  And  so  on. 

In  all  these  respects  we  are  merely  observing  and  describing  in  the 
music  what  can  also  be  observed  and  described  in  the  thoughts,  senti- 
ments, and  activities,  upon  which  we  feel  we  stake  our  whole  personality. 
The  basis  of  affinity  between  the  stuff  of  the  art  in  its  changes  and  our 
personal  mental  life  is  therefore  clearly  real  and  indisputable. 

When  we  feel  that  a  person  is  angry  with  us,  we  do  not  observe 
that  person's  anger,  but  only  his  expressions.  When  we  in  turn  are 
angry  with  any  one,  we  not  only  feel  the  anger,  but  we  express  it  and 
observe  whether  our  expression  of  it  seems  to  us  to  coincide  with  our 
feeUng  as  we  desire  it  to  do.  There  is  an  inner  agreement  between 
the  two  that  satisfies  us.  It  has  often  been  supposed  that  only  at 
this  moment  do  we  become  aware  of  what  anger  means.  That  is 
surely  a  mistake.  It  seems  probable  that  we  could  equally  well  learn 
for  the  first  time  what  anger  is,  in  the  expression  given  to  it  by  some 
other  person.  If  it  were  not  so,  a  savagely  angry  person  might  well 
walk  up  and  finish  us  off  in  our  innocence.  What  less  is  there  to  note 
in  such  a  first-felt  anger  than  in  the  first-expressed  rage  of  our  own? 
Do  we,  this  first  time,  merely  note  how  our  rage  expresses  itself,  being 
entirely  unprepared  for  the  form  it  takes,  and  then  learning  what 
conduct  signifies  rage?  If  our  expression  turned  out  to  be  all  smiles 
and  compliments,  should  we  think  these  were  the  expressions  of  rage? 
I  cannot  conceive  that  any  mind  could  be  built  on  so  silly  and  witless 
foundations.  Such  a  theory  hands  over  all  the  dignity  of  coherence 
to  the  body  and  its  heredity,  making  mind  a  pure  farce.   Why  the  body 


232  AESTHETICS  AS  A  PURE  SCIENCE  [ch.  xxiv 

should  be  held  to  be  thoroughly  subject  to  all-pervasive  law,  while 
the  mind  is  a  mere  epi-phenomenon,  whose  connexions  even  with  the 
body  are  completely  devoid  of  inner  sense  or  of  continuity  with  the 
body,  I  have  not  for  long  been  able  to  appreciate. 

We  can  therefore  in  all  probability  just  as  well  obtain  our  first 
experience  of  anger  from  the  expression  (of  it)  that  a  person  bestows 
upon  us  as  from  the  acts  of  theirs  that  rouse  us  to  an  anger  that  we 
then  proceed  to  express.  We  might,  indeed,  even  encounter  some  emotion 
or  sentiment  for  the  first  time  in  a  work  of  art,  just  because  that  work 
embodied  the  sufficient  basis  of  such  an  emotion. 

From  this  point  of  view  we  can  see  not  only  the  affinity  of  music 
with  our  soul's  life,  but  also  with  other  arts.  The  connexion  with  dancing 
and  all  its  motions  is  obviously  so  close  as  to  be  almost  a  continuity,  a 
complete  fusion.  Where  any  other  visual  display,  e.g.  that  of  the  stage, 
can  keep  properly  in  touch  with  the  speed  of  change  of  music,  there 
can  also  be  an  intimate  union  of  the  two.  But  in  opera  great  concessions 
have  often  to  be  made  to  the  stage,  so  that  a  charge  of  incoherence 
may  here  be  well  founded.  Music  and  sculpture  or  painting  do  not 
cohere  at  all;  for  the  one  is  an  art  of  succession  and  of  motion,  while 
the  others  are  arts  of  simultaneity  and  of  motionless  form ;  and  between 
these  all  true  correlation  of  detail  is  lacking. 


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(68)  DiflFerenztone  und  Konsonanz.  Ztsch.  Psychol.  1905,  39,  269-283. 

(69)   Idem.  2ter  Art.   Ibid.  1911,  59,  161-175. 

(70)  Beobachtimgen  iiber  Kombinationstone.  Ihid.  1910,  50,  1-142. 

(71)  Konsonanz  und  Konkordanz.    Ibid.  1911,  58,  321-355. 

(72)  Die  Attribute  der  Gresichtsempfindungen.  Abh.Preuss.  Ak.Wis8.\9\l  {%). 

(73)  Stumpf,  C.  und  Meyer,  M.  Maasbestimmungen  iiber  die  Reinheit  consonanter 

IntervaUe.    Ibid.  1898,  18,  321^04. 

(74)  TovEY,  D.  F.   Art.  'Harmony.'   Encycl.  Britan^  11th  ed. 

(75)  Art.  'Melody,'  ibid. 

(76)  TsCHAiKOWSKY,  P.    Guide  to  the  Practical  Study  of  Harmony.    Leipzig,  1900. 

(77)  Watt,  H.  J.   The  Psychology  of  Sound.  Cambridge,  1917. 

(78)  Wertheimer,  M.    Experimentelle  Studien  iiber  das  Sehen  von  Bewegungen. 

Ztsch.  Psychol.  1912,  61,  161-265. 

(79)  Westerby,   H.    The  Dual  Theory  in  Harmony.    Proc.   Mus.  Assoc.    1902, 

29,  21-72. 

(80)  Westphal,  R.   Aristoxenv^  von  Tarent.   Leipzig,  1883. 

(81)  WooLDRiDGE,  H.  E.    The  Oxford  History  of  Music.    Vol.  i:  The  Polyphonic 

Period.     Oxford,  1901. 

(82)  Zarlino,  G.     Uistitutioni  harmoniche. 

(83)  Patrologiae  cursus  completus.   Series  Latina  prior.    Ed.  J.  P.  Migne.   Paris. 


INDEX  OF  AUTHORS 


D'Alembert,  J.,  x,  36  f.,  66,  99  f. 
Aristo teles,  quisdam,  186 
Aristotle,  6,  38,  51,  82  f.,  224 
Aristoxenus,  xi,  7,  107,  158 

Bacchiua,  167 

Becker,  E.,  167 

Boethius,  38 

Bridge,  F.  and  Sawyer,  P.  J.,  179f. 

Browne,  W.  D.,  166 

Cherubini,  M.  L,,  92 

Cotto,  J.,  82 

Cross,  C.  R.  and  Goodwin,  H.  M.,  29 

Cummings,  W.  H.,  86 

Curwen,  J.,  39 

Day,  A.,  42,  99,  116fif. 
Debussy,  C.  A.,  16,  140 
Descartes,  R.,  23,  112 
Dunstable,  John  of,  86 

Elert,  K.,  120 
Euclid,  38 
Euler,  L.,  38 

Gardiner,  W.,  12 

Gaudentius,  xi,  15,  108,  157 ff.,  213 

Gevaert,  F.  A.,  38,  51,  52,  81  f.,  108,  146, 

155   157   159 
and'VoU'graff,  J.  C,  xi,  6f.,  52,  82, 

85,  154,  157,  167,  178,  184,  224 
Gladstone,  F.  E.,  84  ff.,  88,  93  ff.,  117f.,  120 
Glyn,  M.  H.,  149 
Guido,  85 

Hauptmann,  M.,  142 
Heffernan,  J.,  24,  33 
Helmholtz,  H.  v.,  vi,  3,  24  ff.,  32  f.,  69  f., 

87ff.   146f.,  162,  176f.,  184,  186ff., 

224 
Holden,  J.,  38,  175 
Hombostel,  E.  v.,  153 
Hucbald,  85 

Hull,  A.  E.,  16,  94,  117,  120,  140,  159,  166 
HuUah,  J.,  63f,,  111 


Jadassohn,  S.,  125 f.,  128 f. 

James,  W.,  69 

Johannes  de  Gariandia,  159,  187 

Kaestner,  G.,  196 
Kemp,  W.,  142,  175 f.,  186 
Kitson,  C.  H.,  124,  126,  128 
Kollmann,  A.  P.  C,  94 
Krueger,  F.,  186 f.,  195 
Kulpe,  0.,  175,  186 

Lasus,  1,  38 
Lee,  v.,  225 
Lipps,  Th.,  26,  38 

Macfarren,  G.  A.,  3,  51,  85 f.,  89,  93,  97 f., 

11 6f.,  133  f.,  136,  166,  175,  192 
Mach,  E.,  100 
Malmberg,  C.  F.,  195  f. 
Maltzew,  C.  v.,  52 f. 
Mansfield,  0.  A.,  117,  135  f. 
Meyer,  M.,  42,  188  f. 
Mies,  P.,  226 
MiUer,  D.  C,  3,  27 
Myers,  C.  S.,  34 

Nicomachus,  38,  154 
Niecks,  P.,  226 

Oettingen,  A.  v.,  142 
Ouseley,  P.  A.  G.,  83,  99 

Parry,  C.  H.  H.,  104,  116  ff.,  162  f. 

Pater,  W.,  225 

Pear,  T.  H.,  175 

Pearsall,  R.  L.  de,  95,  117 

Plato,  83 

Plutarch,  51 

Pole,  W.,  4,84  ff.,  89,  163 

Potter,  A.  G.,  172 

Prout,  E.,  viiif.,  35,  99,  102 ff.,  HI,  116f., 

121  ff.,    128  ff.,   133  ff.,    172  f.,    177 

184 
Ptolemy,  107 
Pythagoras,  38 


INDEX  OF  AUTHORS 


237 


Eameau,  J.  P.,  ixf.,  23,  36 f.,  65 ff.,  79,  99, 

142,  166 
Biemann,  H.,  142 
Rockstro,  W.  S.,  122 
Rousseau,  J.  J.,  39 
Russell,  B.,  100 

Sacchi,  G.,  66 f.,  86,  88 f.,  90ff.,  118f. 

Schubert,  F.,  34  f. 

Scott,  C,  132,  140 

Scriabin,  A.,  16,  80 

Serre,  J.  A.,  28 

Shinn,  F.  G.,  95£E.,  116ff.,  120f.,  125ff., 

136f.,  170f. 
Shirlaw,  M.,  ix,  23  f.,  28,  66 f.,  142,  149, 

182,  184 
Sorge,  G.  A.,  28 
Stainer,  J.,   117 
Stanford,  C.  V.,  162 


Stephens,  C,  85,  89 

StrattoD,  G.  M.,  49  f. 

Stumpf,  C,  xi,  6,  15,  23fif.,  32,  42,  51,  70, 

83f.,  108,  110,  142ff.,  153f.,  156ff., 

164,  184  fE. 

Tartini,  G.,  23,  28,  142,  224  i 

Tchaikovsky,  P.,  106,  116,  123fE.,5l28f., 

135f. 
Thrasyllus,  157 
Tovey,  D.  F.,  54,  129,  151,  164,  182 

Wagener,  A.,  84 

Wertheimer,  M.,  78 

Westerby,  H.,  142 

Westphal,  R.,  7 

Wooldridce,  H.  E.,  51,  82  f.,  92,  159 

Zarlino,  G.,  23,  87,  89,  142 


INDEX  OF  SUBJECTS 


Abstraction,  73  f.,  77,  83,  87 

Aesthetics  as  science,  vi,  98  f.,  131  ff.,  221  ff. 

Analysis  and  synthesis,  Iff.,  55 ff.,  143 f., 

202  f. 
Art  and  science,  22,  55,  60,  64,  101,  139  f., 

150,  214  f. 
Ascent  and  descent  of  pitch,  52,  127,  131, 

184 
Attributes  of  sound,  5ff. 

Balance,  8f.,  19 f.,  57,  72  ff.,  79,  200;  of 

parts,  97;  of  distinction,  147  f. 
Bass  most  prominent,  51,  92,  102,  130 
Beats,  24  ff.,  26  f.,  31,  56,  180  f. 
Beauty  objective,  216  ff. 
Blend,  5,56  ff. 
Breadth  of  tone,  7,  13  f. 

Chords,  theory  of,  148  ff.,  163  ff.,  209  ff. 

Cochlea,  46  f.,  203  f. 

Columns  of  thirds,  etc.,  79  f.,  148  ff.,  163, 
209  f. 

Concordance,  141  ff. 

Consecutives,  theories  about,  81  ff.;  facts 
about,  101  ff . ;  explanation  of  pro- 
hibition, 109 ff.;  exceptions  to  pro- 
hibition, 115 ff.;  171  f.,  207 

Consonance,  grades  of,  15  ff.,  83,  94,  184  ff., 
223 

Consonance  and  dissonance,  15 ff.;  Helm- 
holtz's  theory  criticised,  24  ff.; 
Stumpf's  theory  criticised,  146 ff.; 
151  ff.,  155;  in  successive  tones,  25, 
31ff.,  33ff. 

Convention  or  habit  in  music,  84  ff.,  89, 93  f., 
96,  132,  216  f.,  230 

Development  in  music,  15 f.,  62,  92,  96, 113, 
150,  152  ff.,  159,  161  ff.,  168,  196  f., 
212f. 

Difference  tones,  28  ff.,  31,  56,  58  f.,  192  f., 
196,  202,  224 

Distinction,  loss  of,  147  f . 

Ear,  training,  179  f.;  absolute,  189;  verdict 
of,  204 


Exposed     intervals,     v.     hidden,     125  f., 

136  f. 
Exposure,  rhythmical,  134,  137  f. 
Expression  in  music,  229  ff. 

Factors,  positive  and  negative,  in  beauty, 
89,  96,  127  f.,  173  ff. 

Field,  the  auditory,  48  ff.,  53  f.,  201  f. 

Fifth,  the,  19  f.,  29,  33  f.,  59;  a  bare  interval, 
83,  96;  fifths  not  equivalent,  77 ff.; 
consecutive,  80  ff. 

Form  and  mass  in  hearing,  43  f . 

Fourth,  the,  21,  133  ff.,  177  f.,  191  f.,  210; 
as  dissonance,  136f. 

Fusion,  11,  15 ff.,  18  ff.,  33  ff.,  43,  62  f.,  64, 
74,  78,  94,  96,  173 f.,  190;  explana- 
tory value  of,  141  ff.,  185 ff.;  grades 
of,  16  f.,  21  ff.,  32,  71,  102  ff.,  llOff., 
122  ff.,  147  ff.,  151  ff.,  156,  184  ff., 
200 f.;  neutral  grades  of,  108,  lUf., 
147  f.,  187,  195 

Greek  music  and  theory,  6f.,  15,  18,  51, 
81  ff.,  92,  107  f.,  110,  146,  151,  154 ff., 
167, 178, 185,  206,  213 

Harmonics,  v.  partials 

Hidden  intervals,  122 ff.;  v.  exposed 

Highest  tones,  12,  45  ff. 

Horizontal  and  perpendicular,  63 f..  Ill  f. 

Imagination,  force  of,  90  f. 

Individual  differences  in  ear,  61 

Instrumental  tones,  1  ff. 

Intensity,  6,  8,  12,  19,  72,  198  f. 

Interval,  17,  36  ff.,  200;  in  inversions,  71  f., 
173 ff.;  cause  of  precision  of,  192 f.; 
nomenclature  of,  16;  ascending  and 
descending,  52  f.;  simultaneous  and 
successive,  53;  intrinsic  character, 
95  ff. 

Inversion  of  chords,  66  ff.,  76  f.,  88,  95,  149, 
151  f.,  175ff.,  203,  209 

Longitudinal  and  transverse,  13  f. 
Lowest  tones,  12,  45  ff. 


INDEX  OF  SUBJECTS 


239 


Major  and  minor  triads,  difference  between, 
52,  142,  166  f. 

Materialistic  theories,  prejudice  for,  v,  204 

Melody,  205;  statistics  of,  34  f.;  in  bass 
voice,  51  f. ;  as  surface  of  music,  54, 
61;  as  basis  of  prohibition  of  con- 
secutives,  lllff.,  126;  of  hidden  or 
exposed  intervals,  128ff. ;  of  fourth 
from  bass,  134ff. 

Melodic  distinction,  148,  156,  160  ff.;  flow, 
172,  178,  182,  206,  208  f.;  steps  and 
leaps,  34 f.,  124 f.,  128  ff.,  134  ff. 

Memory,  27  f.,  31,  42,  194,  196 

Method,  101  f.,  109  f.,  122 

Motion,  contrary  and  similar,  82,  170  ff., 
207  f. 

Mystery  in  music,  140 

Objectivity  of  beauty,  214  ff. 

Octave,  the,  19,  33,  42  f.,  59;  equivalence 

of,  65ff.,  96,  113  f. 
Order  or  position,  7,  48  ff.,  198  f. 
Organum,  82  f. 
Overlapping  of  tones,  10  ff.,  18  ff. 

Paraphony,  155 ff.,  187 ff.,  206 ff.,  230 ff.; 
as  basis  of  polyphonic  music,  160 ff.; 
factors  that  modify,  169  ff.;  inter- 
action of,  173  ff . 

Partial  tones,  viii,  1  ff.,  15  f.,  149,  166  f., 
180 f.,  192  ff.,  196,  202;  relation  to 
fusion,  25  ff.,  31;  in  synthesis  and 
analvsis,  56  ff.,  61,  67  ff.,  76  f .,  87  ff. 

Particles  of  sensation,  50;  of  sound,  9  ff.. 
18  ff. 

Parts,  the  number  of,  130  f . 

Pattern,  72ff„  145,  165,  203,  211 

Pitch,  4,  7ff.;  rising  as  departure,  52; 
musical  range  explained,  45  ff.,  de- 
fined, 47;  relation  to  volume,  8ff. ; 
central  in  volume,  9ff.,  19,  199 

Pleasure  in  relation  to  beauty,  217 

Polyphony  and  harmony,  43 f.,  62 ff..  Ill, 
162  ff.,  205 

Primary  tones  in  analysis  and  synthesis,  60  ff . 

Proportion,  36  f.,  73 

Pure  psychology,  5, 20  ff.,  37, 54, 98  ff.,221  ff. 

Pure  tones,  3ff. 


Quality,  5;  amongst  the  tones  of  an  octave, 
70  f.,  78 

Relationship  of  chords,  94  ff.,  117f.,  120 f., 

127  ff.,  184,  212 
Representation  in  music,  etc  225  ff . 
Resolution,  137,  182  f.,  211 
Rest  and  motion,  musical,  151  ff.,  155 
Rhythm,  91,  137,  178,  230 
Rules  in  music,  81,  84  ff.,   132.  216  f.,  v. 

convention 

Sensations,  5 

Seventh,  diminished,  106 

Sixth  minor  and  augmented  fifth,  180  ff.,  210 

Smoothness  in  tone,  8f.,  25,  57 

Sonorous  body,  the,  ix,  66  f.,  149 

Statistics,  viii,  34 f.,  115,  117  ff.,  139  f. 

Subjectivism,  214  ff. 

Surface  of  tone,  57  f. 

Symmetrical,  8f. 

Synergy,  23  f.,  32  f.,  70 

Thirds  and  sixths  as  consonances,   107  f., 

142,  147  f.,  154,  157  ff. 
Tonality,  53,  90ff.,  164 ff.,  183 f.,  211  f. 
Tones  and  noises,  8 
Toiros,  7 
Trend    of    opinion,    104,    115,    123,    128, 

131  f. 
Triad,  theory  of,  148  ff. 
Tritone,  103',  105  f. 

Unconscious,  the,  59 
Unison,  the,  130,  151 

Vision,  compared  with  hearing,  33,  40  f., 
43,  48  ff.,  73,  85 

Voices,  relative  prominence  of,  51, 92, 102  ff., 
122  ff.,  128 f.,  208;  effect  of  number 
of,  127 f.,  169f.,  208;  llOf. 

Volume,  6ff. ;  inversely  proportional  to 
ratio  of  vibration,  20;  graded  in  in- 
tensity, 72,  198  f. 

Western  music,  early,  51,  82  f.,  151 

Zones,  93  f.;  of  paraphony,  etc.,  188  ff. 


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